Kvantové zpracování informace s laditelným hradlem pro kontrolovanou změnu fáze

Transcription

1 PŘÍRODOVĚDECKÁ FAKULTA UNIVERZITY PALACKÉHO V OLOMOUCI SPOLEČNÁ LABORATOŘ OPTIKY Kvantové zpracování informace s laditelným hradlem pro kontrolovanou změnu fáze HABILITAČNÍ PRÁCE Karel Lemr, Ph.D. OLOMOUC DUBEN 016

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3 FACULTY OF SCIENCE, PALACKÝ UNIVERSITY IN OLOMOUC JOINT LABORATORY OF OPTICS Quantum information processing with the tunable controlled-phase gate HABILITATION THESIS Karel Lemr, Ph.D. OLOMOUC APRIL 016

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5 v Declaration of originality I hereby declare that this thesis is my own work and that, to the best of my knowledge and belief, it contains no material previously published or written by another person nor material which to a substantial extent has been accepted for the award of any other degree or diploma of the university or other institute of higher learning, except where due acknowledgement has been made in the text. Olomouc, 5th April 016 Digital signature: Karel Lemr (k.lemr@upol.cz)... Submitted The author grants permission to Palacký University in Olomouc to store and display this thesis and its electronic version in university library and on official website.

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7 vii Acknowledgement I wish to express my sincere gratitude to all my colleagues, most notably to (in alphabetical order) Dr. Karol Bartkiewicz, Dr. Antonín Černoch, prof. Miloslav Dušek, Dr. Ondřej Haderka, and Dr. Jan Soubusta. My thanks also belong to all co-authors of my publications. I am most thankful to my family, especially to my wife Barborka and son Karel, to whom I dedicate this work. The author Light thinks it travels faster than anything but it is wrong. No matter how fast light travels, it finds the darkness has always got there first, and is waiting for it. Terry Pratchett, Reaper Man

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9 Contents Foreword xi 1 Going quantum Quantum information processing QIP with linear optics and individual photons Tunable linear-optical controlled-phase gate 7.1 Principle of operation Linear-optical implementations Our experimental implementation Properties of the tunable controlled-phase gate Success probability of the controlled-phase gate Entangling efficiency of linear-optical quantum gates Applications of the tunable controlled-phase gate Preparation of Knill-Laflamme-Milburn states Linear-optical quantum routers Applications of quantum routers Implementation of controlled-unitary gates Programmable controlled-phase gate 39 6 Linear-optical qubit amplifiers Entanglement-based linear-optical qubit amplifier State-dependent linear-optical qubit amplifier Conclusions Future plans Author s publications 53 References 55 ix

10 x CONTENTS Supplementary Material S-1 Optimal success probability of a tunable linear-optical controlled-phase gate S-3 Entangling efficiency of linear-optical quantum gates S-7 Preparation of Knill Laflamme Milburn states using a tunable controlled phase gate S-13 Linear-optical programmable quantum router S-19 Resource-efficient linear-optical quantum router S-3 Using quantum routers to implement quantum message authentication and Bell-state manipulation S-31 Experimental Implementation of Optimal Linear-Optical Controlled-Unitary Gates S-41 Scheme for a linear-optical controlled-phase gate with programmable phase shift S-47 Entanglement-based linear-optical qubit amplifier S-53 State-dependent linear-optical qubit amplifier S-61

11 Foreword Quantum computation and communications is a promising scientific field, in which the information sciences merge with quantum theory. Application of quantum phenomena to information processing results in more efficient computation algorithms or inherently secure communications. Individual photons are one of the suitable platforms for theoretical and experimental research in this field. Because of its properties, light is suitable namely for testing of fundamental concepts and for practical quantum communications. In this thesis, the author comments on ten scientific papers published by him and his co-workers over the course of five years [A1 A10]. Presented publications report on original research in the field of quantum information processing with discrete photons. In some cases, this research involves theoretical analyses, while in other cases experimental results are included as well. The applicant, who is simultaneously the corresponding author of all the commented publications, received in 013 funding by the Czech Science Foundation dedicated specifically to these research activities. This work focuses mainly on the properties and applications of the tunable controlled-phase gate, which happens to be a key ingredient of many quantum information protocols. Research has revealed how to use this gate for efficient quantum operations or in quantum communications tasks. Subsequently, this thesis comments on a closely related topic of qubit amplification. The concept of entanglement-based qubit amplifier is discussed and its benefits explained. Results presented in this thesis contribute to the ongoing research in the field of quantum information processing. Some of the ideas allow to increase efficiency of quantum computing, while other put forward useful devices for quantum communications. As a whole, the obtained results constitute steps towards future deployment of practical quantum communications networks. xi

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13 Chapter 1 Going quantum It was the Fourteenth of December, 1900 and the old century was about to end in two weeks. On this very day a German physicist Max Planck stood up in front of his colleagues from the German Physical Society and presented his theory of black body radiation [1] 1. As a purely formal assumption he postulated that transfer of energy between matter and light is not continuous, but rather quantised []. This idea marked a new era for the entire physical science and it is only fitting that is was presented at the dawn of a new century. Soon afterwards, Planck s idea attracted interest of the most brilliant scientists of the beginning of the 0th century. Physicists such as Niels Bohr, Max Born, Louis de Broglie, Paul Adrien Maurice Dirac, Albert Einstein, Werner Karl Heisenberg, Wolfgang Pauli, Erwin Rudolf Josef Schrödinger, and Jára Cimrman laid foundations of a new field of science called quantum physics [3]. Since its conception, quantum physics managed to predict and correctly describe a number of natural phenomena some of which, like quantum entanglement, seemingly defy the common sense [4, 5]. Experiments conducted in laboratories have so far proven quantum models correct even in situations where it contradicts classical physics [6, 7]. Having a powerful model to understand the universe is most enlightening, but using this model to make our lives easier is even better. Quantum physics has the potential to enhance a number of classical technologies. Derived fields of science searching for ways of how to use quantum phenomena for such enhancement have spawned over the course of the 0th century. In quantum metrology, quantum physics allows to increase measurement precision beyond its classical limit [8]. In quantum lithography, various quantum states of light provide better tools to work with then classical light beams [9]. There are even quantum games promising more fun than their classical counterparts [10]. 1 The German Physical Society actually publishes on their website scanned protocol from that particular meeting. 1

14 CHAPTER 1. GOING QUANTUM 1.1 Quantum information processing Quantum physics has also penetrated into the information sciences resulting in a new discipline called the quantum information processing (QIP) [11 13]. Quantum physics brings to this field of science its peculiar phenomena such as the principle of superposition and entanglement. The information is no longer stored in classical bits with logical values of 0 and 1. Quantum bits, called qubits, are used instead [14]. Quantum-physical laws allow them to be in any superposition of both the logical states 0 and 1. The benefit of replacing classical bits with qubits becomes apparent when one considers that a general superposition of logical states 0 and 1 is specified by two real numbers θ and φ ψ = cos θ 0 + eiφ sin θ 1 (1.1) (θ denoting amplitude ratio between 0 and 1, φ their mutual phase shift). These real numbers can, in principle, be inscribed into the qubit with infinite precision and therefore unlimited amount of classical information can be stored within one single qubit. When such qubit is inserted into a quantum gate, all the information it bears is processed simultaneously. Unfortunately, subsequent measurement allows to extract only one bit of information from one qubit. Nevertheless this limitation does not prevent quantum computing from increasing efficiency of various algorithms such as database searching (Grover s algorithm [16]) or factorising products of two unknown prime numbers (Shor s algorithm [17]). Quantum information protocols benefit also from other non-classical features such as the quantum entanglement [18] or the impossibility to copy an unknown quantum state 3. Although the majority of quantum information research is still in the stage of theoretical investigation and laboratory testing, one notable exception of an already applied QIP technology is the quantum cryptography (see Fig. 1.1) [0]. In contrast to classical cryptography, where security is based on computationally difficult decryption, quantum cryptography guarantees security by the laws of physics. Experimentalists have achieved encoding and processing of quantum information on various physical systems such as individual photons [1], coherent light beams [], individual atoms [3], collective atomic states [4], nuclear magnetic resonance [5] or quantum dots [6]. In order to encode a qubit on any physical system, one has to find a suitable degree of freedom that supports at least two orthogonal states ( living thus in a two-dimensional Hilbert space). While the solid-state platforms seam to be more suitable for construction of quantum memories, light, on the other hand, is a prominent choice for quantum communications networks. Dirac bracket formalism is used to denote quantum states [15]. 3 The no-cloning theorem in quantum physics states that it is impossible to perfectly duplicate an unknown quantum state (e.g. a qubit) [19].

15 1.. QIP WITH LINEAR OPTICS AND INDIVIDUAL PHOTONS 3 Figure 1.1: Cerberis quantum key distribution server by ID Quantique is an example of quantum information technology that has already reached the production stage. [Photo by ID Quantique jpg] 1. Quantum information processing with linear optics and individual photons As mentioned in the preceding section, individual photons are one of the suitable physical platforms for quantum information processing [1]. Even one single photon has various degrees of freedom that can be used to encode qubits. Propagating electromagnetic wave, including a single photon, can be characterised by its polarisation geometry of oscillations of the electric field vector in the plane transversal to the propagation direction (see Fig. 1.a) [7]. Since the transversal plane is obviously two-dimensional, one can always find two mutually orthogonal polarisation states (e.g. horizontal and vertical linear polarisation). These two states form a basis in two-dimensional Hilbert space straightforwardly suitable for qubit encoding. In the majority of the author s papers, as well as in a number of other articles, this particular qubit encoding is preferred, namely because of its experimental user friendliness. In the laboratory, we use series of half and quarter-wave plates to arbitrarily manipulate the photons polarisation state. Mathematically, we assign horizontal polarisation to the logical state 0 and vertical polarisation to the logical state 1. Any complex superposition of logical states 0 and 1 is then represented by a specific polarisation state. For instance the logical state 1 ( ) is encoded as a diagonally polarised photon while the state 1 ( 0 + i 1 ) corresponds to right-hand circular polarisation. It is convenient to visualise various polarisation states using the Bloch sphere (see Fig. 1.b). Logical state 0 and 1 (horizontal and vertical polarisation) are situated

16 4 CHAPTER 1. GOING QUANTUM (a) (b) Figure 1.: (a) Horizontally and vertically polarised electromagnetic wave. (b) Convenient way to visualise various polarisation states and their mutual relations using the Bloch sphere. Polarisation states are labelled as follows: H horizontal, V vertical, D diagonal, A anti-diagonal, R right-hand circular, L left-hand circular. Figure 1.3: Spatial (dual-rail) qubit encoding. Two spatial modes (implemented by fibres) are designated to be logical states 0 and 1. By means of a fibre coupler and stretcher, a photon in original logical state 0 is transformed into a general qubit state of the form of (1.1) propagating in superposition of both spatial modes. on the poles, while their balanced superpositions populate the equator. When using fibre optics, it seams more convenient to use spatial qubit encoding (also known as dual-rail encoding). In this case we designate one propagation mode (e.g. one fibre) to be the logical state 0 and a second mode to be the state 1 (see Fig. 1.3). Quantum superposition allows for the photon to propagate simultaneously in both these modes resulting again in a valid qubit encoding. When encoding a specific qubit in the form of (1.1), one usually uses a fibre coupler (fibre version of a beam splitter) to split the wave packet of the photon between the modes according to the ratio given by tan θ. The relative phase shift φ between the modes is then implemented using for instance a fibre stretcher. In free space, it is more difficult to maintain phase stability between two spatial modes then to maintain the photon s polarisation state. Spatial mode encoding is thus used mostly with polarisation non-maintaining optical components such as ordinary fibres. There are also other possible qubit encodings besides the ones discussed above.

17 1.. QIP WITH LINEAR OPTICS AND INDIVIDUAL PHOTONS 5 - (a) (b) Figure 1.4: (a) Schematic representation of an ideal beam splitter parametrised by intensity transmissivity T. Reflection of one mode is phase shifted by π because of Fresnel relations. Annihilation operators of its output modes are linear combinations of annihilation operators of its input modes as prescribed in Eqs. (1.) and (1.3). (b) Two-photon interference (Hong-Ou-Mandel effect) occurring on a balanced beam splitter. Two photons in individual input modes bunch to a common output mode. As a result, the output state is a superposition of two photons in the first output mode and two photons in the second. Experimentalists have successfully tested for instance time-bin encoding [8] or orbital angular momenta encoding [9], but neither of these has been used in any of the author s papers. In order to process quantum information encoded into a photon s state, we subject this photon to various linear-optical components. An optical component is said to be linear if annihilation operators of its output modes are linear combinations of annihilation operators of its input modes [30]. Such relation can thus be conveniently expressed in a matrix form â OUT = V â IN, (1.) where â IN and â OUT stand for vectors of annihilation operators of input and output modes. V then denotes matrix form of the transformation imposed by the linear-optical component. As an example, ideal beam splitter with intensity transmissivity T performs a two-mode linear transformation with the matrix T 1 T V = 1 T T (1.3) (see Fig. 1.4a) [31]. Other linear-optical components include polarising beam splitters, wave plates, phase shifters and various similar fibre optical tools. All these components can be described by transformation in the form of (1.) with their own specific forms of the V matrices. The underlying phenomenon behind linear-optical quantum information processing is the interference, either of a photon with itself (single-photon interference) or with another photon (two-photon interference) [1]. Single-photon in-

18 6 CHAPTER 1. GOING QUANTUM terference occurs when two modes (e.g. polarisation or spatial), in which the photon is propagating, are coherently mixed. It allows to implement various singlequbit transformations and in the laboratory single-photon interference closely resembles interference of classical light beams. The two-photon interference, on the other hand, is purely a quantum effect. It is the core of information processing in linear-optical quantum logical gates. To illustrate this phenomenon, imagine two photons impinging on a balanced beam splitter (partially reflective mirror with equal transmissivity and reflectivity) as depicted in Fig 1.4b. The four possible outcomes are: (i) both photons are transmitted, (ii) both photons are reflected, (iii) the first photon is transmitted while the other reflected and finally (iv) the first photon is reflected and the second transmitted. In terms of photon numbers, this transformation can be expressed as , (1.4) }{{}}{{} (i) (ii) }{{}} {{} (iii) (iv) where the numbers in brackets denote the numbers of photons in the first and second mode respectively. Note that the cases (i) and (ii) result both in one photon in each output mode. If these photons are completely indistinguishable in all degrees of freedom (such as polarisation, time of arrival, transversal profile,...) these two cases lead to an identical outcome. The fact that the phase is shifted by π when the photons get reflected causes these two outcomes to mutually cancel each other. As a result, the photons bunch to one of the output modes with zero probability of them to travel separately. This result is also known as the Hong- Ou-Mandel effect [3]. When designing linear-optical quantum gates, the parameters of all the components are suitably selected so that the single and two-photon interference occur in a desired way. It has been shown that linear-optical components alone are not sufficient to construct all quantum gates necessary for universal quantum computing [33, 34]. One can resort to non-linear optical components such as Kerr media [35]. This option is however experimentally very challenging if not impossible since the non-linear interaction between two photons is extremely weak [36]. Alternatively, one can combine linear-optical tools with additional ancillary modes (both vacuum or populated). Measurement on these modes induces the non-linearity needed for construction of quantum gates [37]. Gates built this way however succeed only probabilistically, when this measurement yields a specific subclass of potential outcomes. For instance, each photon entering the setup must leave it by a separate designated output port. Cases when two photons take the same output port or a photon leaks into an incorrect mode are discarded. As a result, some quantum gates, like the controlled-phase gate, can only be implemented probabilistically with linear optics. This fact imposes limitations to their scalability and efficiency. It is therefore crucial to search for solutions maximising the success probability of the gates.

19 Chapter Tunable linear-optical controlled-phase gate Similarly to classical electronics, quantum circuits can be decomposed to a number of elementary building blocks. It has been shown that a relatively small set of single and two-qubit quantum gates suffices for construction of any quantum circuit [38]. This set includes several single-qubit gates and only one two-qubit gate the controlled-phase gate (or alternatively the controlled-not gate). While the single-qubit gates can easily be implemented (even deterministically with linear optics) 1, the controlled-phase gate is more challenging and definitely non-trivial..1 Principle of operation The controlled-phase (c-phase) gate operates on two qubits designated as control qubit and target (or signal ) qubit. Transformation imposed by this gate takes the form of 00 00, 01 01, 10 10, 11 e iϕ 11, (.1) where the numbers in brackets denote logical states of the target and control qubits respectively and ϕ stands for the phase shift added to these qubits if and only if both are in logical state 1 [1]. The c-phase gate relies on an interaction between the qubits as visualised in the conceptual scheme in Fig..1. When the two qubits are in logical states 1, 1 Here we assume that one qubit is encoded into the state of one photon. With respect to the controlled-phase gate, the target and signal qubits are synonyms and are used interchangeably throughout this text. 7

20 8 CHAPTER. TUNABLE LINEAR-OPTICAL CONTROLLED-PHASE GATE Figure.1: Conceptual scheme of the controlled-phase gate. The gate imposes a phase shift ϕ when both the qubits are in logical states 1. In all other cases, the qubits do not interact and their state remains unchanged. = (a) (b) Figure.: (a) Symbolic representation of the c-phase gate used in quantum circuits design. (b) Symbolic representation of the CNOT gate and its decomposition to a controlledsign gate enveloped by two Hadamard transforms acting on the target qubit. they interact inside the gate and, as a result, a phase shift ϕ is appended to their state [37]. Otherwise no interaction takes place and no phase shift is added. In general, the c-phase gate can impose any phase shift ϕ in the range [0; π]. In quantum circuit design, the c-phase gate is usually denoted by the symbol shown in Fig..a. Special case of ϕ = π is however sufficient for universal circuit design [38]. This setting of the c-phase gate is also known as the controlled-sign gate since e iπ = 1 and thus the gate performs a conditional sign flip [37]. The controlledsign gate is also closely related to the controlled-not (CNOT) gate [1]. As illustrated in Fig..b, the controlled-not gate is obtained by simply overlapping the controlled-sign gate by two Hadamard gates operating on the target qubit. Hadamard gates implement single-qubit transformation 0 1 ( ), 1 1 ( 0 1 ) (.) which together with the controlled-sign gate gives the CNOT transformation of

21 .. LINEAR-OPTICAL IMPLEMENTATIONS 9 the form of 00 00, 01 11, 10 10, (.3) The target state NOT inversion is performed, when the control qubit is in logical state 1, otherwise the gate has no effect on the target state. One of the key properties of the c-phase gate is its entangling capability [39]. To demonstrate this aspect of the gate, let us assume both target and control qubits to be prepared in balanced superpositions of the logical states, that is ψ t = ψ c = 1 ( ). (.4) The overall state of both the target and control qubits thus reads ψ t ψ c = 1 ( ). (.5) One can immediately calculate, how the c-phase gate transforms this separable state ψ t ψ c e iϕ 11. (.6) From the equation (.6) it follows, that for all values of ϕ other than multiples of π, the output state is no longer separable, but entangled [39]. Specifically for ϕ = π we obtain a maximally entangled Bell state. This entangling property of the c-phase gate proves particularly useful as it will be discussed Chapter 3. Given the importance of the c-phase gate (or CNOT gate), it is hardly surprising that it has been investigated both theoretically and experimentally on various physical platforms. In 1995, Q. A. Turchette and co-workers have demonstrated the first c-phase gate on light interacting with atoms inside a resonator [40]. Subsequently, D. Cory et al. have achieved its implementation using nuclear magnetic resonance [41]. F. Schmidt-Kaler et al. then reported on a controlled-phase gate using trapped ions [4] and J. H. Plantenberg et al. demonstrated the gate on superconducting qubits [43].. Linear-optical implementations As shown in the conceptual scheme (Fig..1), the phase shift ϕ is imposed only when both the target and control qubits interact. In order to implement such interaction using discrete photons, one would need a medium with the following

22 10 CHAPTER. TUNABLE LINEAR-OPTICAL CONTROLLED-PHASE GATE Figure.3: Experimental scheme used for construction of the controlled-sign gate by O Brien et al. [39]. Polarisation encoding of input qubits is selected so that when both the target and control qubits are in logical state 1, they meet in the same spatial mode and interact. Two beam displacers (BD) are used to transform polarisation modes into spatial modes and back again. Inside the beam displacers, horizontally polarised light travels in its original direction while the vertically polarised beam is displaced (see white flashes in top left corner of the first BD). A half-wave plate (HWP) is inserted in between the BDs to mix polarisation modes and thus allow the photons to interfere. properties: 0 0, 1 1, e iϕ, (.7) where numbers denote the numbers of photons travelling through that medium [37]. The first and second equation state that no phase shift is added when zero or only one photon travels through the medium. The third equation, on the other hand, describes the appended phase shift in the case of two photons simultaneously propagating through the medium. This transformation is clearly non-linear in annihilation operators and therefore the c-phase gate can not be implemented by any set of linear-optical components alone [1]. One thus has to resort to non-linear optical tools, or alternatively induce non-linear effect by the process of measurement and post-selection. The idea of combining linear-optical tools with measurement-based post-selection was presented in 001 in a seminal paper by Knill, Laflamme and Milburn (KLM) [37]. These authors have shown how to implement the non-linear interaction between photons using only linear-optical components and ancillary photons that are subsequently measured. The resulting gate is however necessarily non-deterministic, an inherent property of the linearoptical platform. In 003, O Brien et al. demonstrated a linear-optical implementation of the controlled-sign gate motivated by the KLM approach [39]. Instead of using addi-

23 .. LINEAR-OPTICAL IMPLEMENTATIONS 11 Figure.4: Polarisation dependent beam splitter (PDBS) as a key building block of the controlled-sign gate [44]. Transmissivity for horizontal polarisation (logical state 0 ) is T H = 1 and thus the horizontally polarised photons just pass through and do not interact. On the other hand, transmissivity for vertical polarisation (logical state 1 ) is T V = 1/3 and thus the vertically polarised photons interact. Note that the photons do not interact if one is horizontally polarised and the other vertically polarised, since in that case they are in mutually orthogonal (polarisation) modes. tional photon ancillae, they performed detection on the target and control photons themselves. Thus reducing considerably the resources needed at the expense of collapsing the output state. Their experimental setup consisted of a block of birefringent material (beam displacer) in which the horizontal and vertical polarisation modes of a photon split into two spatial modes subsequently propagating parallel to each other (see Fig..3). This way the experimentalists were able to achieve that the target and control photons, when in logical states 1, shared the same spatial mode. Subsequent linear transformation on all modes was imposed by a half-wave plate resulting in the phase shift π added when both the target and control qubits shared a spatial mode. A second beam displacer was used to recombine the spatially split polarisation modes back to one spatial mode for each photon. With probability of 1/9, each photon leaves the setup by its designated output port and the gate succeeds. Note that the success probability of 1/9 is also the theoretical limit for linear-optical implementation without any additional ancillary photons. The reported fidelity with which the gates processed logical basis states ( 00, 01, 10, 11 ) was about 80% in average. Soon afterwards, it has been realised that the controlled-sign gate can be implemented using only one polarisation dependent beam splitter instead of the above mentioned scheme (see Fig..4). Let us consider logical state 0 being encoded into horizontal polarisation while the logical state 1 into vertical polarisation. Then the optimal parameters of the beam splitter implementing the c-phase gate are: T H = 1 (transmissivity for horizontal polarisation) and T V = 1/3 (transmissivity for vertical polarisation) 3. The target and control photons only interact on this beam splitter if they are vertically polarised. In this case, the beam splitter imposes a π phase shift. In 005, Kiesel et al. implemented the controlled-sign 3 Assuming an ideal lossless beam splitter, the corresponding reflectivity for vertically polarised light is R V = 1 T V = /3.

24 1 CHAPTER. TUNABLE LINEAR-OPTICAL CONTROLLED-PHASE GATE Figure.5: Conceptual scheme of the Lanyon et al. implementation of the tunable controlled-phase gate [46]. If in logical state 1, the target qubit is transferred to an auxiliary mode where it interacts with the control qubit in a CNOT gate. This CNOT gate is implemented using a polarisation dependent beam splitter similar to the Kiesel et al. approach [44]. Depending on the state of the control qubit, the target qubit gets either negated to logical state 0 or not (remains 1 ). Subsequently an unconditional phase shift ϕ is applied to the logical state 0 of the target qubit. Finally, a Hadamard transform is used together with a second polarising beam splitter to recombine the auxiliary and original target qubit modes. gate based on such partially polarising beam splitter [44]. Their experimental setup made use of only one two-photon interaction and did not require stabilisation with interferometric precision (fractions of wavelength). The experimental simplification allowed to reach better stability and thus also higher fidelity while preserving the optimal success rate of 1/9. Successful outcomes are, as usual, post-selected by observing each photon in its designated output mode. Further to that, this method is also suitable to implement the CNOT gate on the platform of integrated optics [45]. All the implementations mentioned so far have only investigated the gate with phase shift ϕ = π. The first controlled-phase gate implementing a different phase shift was demonstrated by B. P. Lanyon and is colleagues in 009 [46]. The conceptual scheme of their approach is depicted in Fig..5. In order to implement a tunable phase shift, the scheme uses an additional auxiliary mode. In fact, they have implemented the c-phase gate by combining the CNOT gate with an unconditional phase shift. The target qubit, only if in logical state 1, is transferred to an spatial auxiliary mode. There it interacts with the control qubit in a CNOT gate. Depending on the state of the control qubit, the target qubit is either kept in the state 1 (control qubit being 0 ) or is negated and becomes 0 (control qubit being 1 ). Subsequently, an unconditional phase shift 0 e iϕ 0, 1 1 (.8) is implemented on the target qubit. In the next step, the target qubit enters a Hadamard gate followed by a filter letting only the logical state 1 pass. As a result, the phase shift ϕ is added only if the target qubit as well as the control qubit

25 .3. OUR EXPERIMENTAL IMPLEMENTATION 13 are in logical state 1. It should however be pointed out that this implementation is non-optimal even for the phase shift ϕ = π. It allows to attain success probability of 1/1 at best..3 Our experimental implementation In 010, K. Kieling et al. [47] found optimal success probability of the controlledphase gate as function of its phase shift ϕ considering only linear optics and no additional photon ancillae P CNOT (ϕ) = 1 + sin ϕ π ϕ + 3/ sin ϕ sin 4 1/. (.9) While it was already known that P CNOT (π) = 1/9 [39, 44], values of this function for other phase shifts ϕ were a newly established fact. Most interestingly, the success probability function is non-monotonic and reaches a minimum value of about 1/11 at ϕ π/3. K. Kieling and his colleagues proposed only a conceptual scheme to implement the tunable c-phase gate with such optimal success probability. In collaboration with Konrad Kieling and Jens Eisert, we have experimentally implemented this scheme and tested its functionality for various values of ϕ [A11]. The experimental setup is depicted in Fig..6. Both the target and control qubits were encoded into polarisation states of individual photons. These photons were generated in the process of spontaneous parametric down-conversion in a LiIO 3 crystal pumped by 50 mw of continuous laser beam. The core of our experimental setup are two nested Mach-Zehnder interferometers. For the construction of the outer interferometer, we used two polarising beam splitters. The inner interferometer is then built using two blocks of birefringent material (beam displacers). The outer interferometer spatially separates photons of logical states 0 and 1 of both the control and target qubits. Photons in logical states 0 propagate by its lower arm, where they do not interact. On the other hand, photons in logical states 1 enter the upper arm of the outer interferometer. In this arm, an inner interferometer is inserted. Half-wave plates situated in front and behind the inner interferometer mix horizontal and vertical polarisation modes and allow the photons to interact in the inner interferometer. One can tune the phase shift between the arms of the inner interferometer as well as losses added to one of its arms. By specific setting of these parameters, any phase shift from 0 to π can be achieved. The gate succeeds when both the control and target photons leave the setup by their designated output ports. In order to fully characterise our gate, we have performed a complete process tomography for various values of phase shift ϕ. These tomographic data allowed us to reconstruct process matrices describing the implemented c-phase operation as a function of the phase shift ϕ. Fidelities of these process matrices were typically about 90% and the experimentally determined success probability was close

26 14 CHAPTER. TUNABLE LINEAR-OPTICAL CONTROLLED-PHASE GATE 1 MT1 HWP PBS1 QWP HWP BDA F1 QWP BDA1 F HWP1 HWP11 piezo MT F1 HWP PBS HWP1 DV D1V HWP QWP QWP HWP PBS PBS D1H DH BDA = BD BD HWP@45deg piezo Figure.6: Experimental scheme for the construction of the optimal tunable linearoptical c-phase gate. Polarisation encoding of target and control qubits is selected so that photons in logical states 0 pass by the lower arm of the outer Mach-Zehnder interferometer (formed by PBS1 and PBS), where they do not interact. Photons in logical states 1 interfere in the small inner interferometer placed in the upper arm of the outer one. Individual components are labelled as follows: HWP and QWP half and quarter-wave plates, F neutral density filters, BD beam displacers, BDA beam displacer assembly (two BDs), D detectors. Motorised translations (MT) are used to stabilise the two-photon overlap and balanced lengths of the outer interferometer arms. Piezo-driven translations are used for interferometric stabilisation (single-photon interference). Figure.7: Success probability of the optimal linear-optical c-phase gate as function of its phase shift ϕ. Note the non-monotonic behaviour of this function having a minimum value of about 1/11 close to ϕ = π/3.

27 .3. OUR EXPERIMENTAL IMPLEMENTATION 15 to its theoretical prediction. We were thus able to verify the counterintuitive fact that the success probability has a non-monotonic dependence of the phase shift ϕ (see Fig..7). The uniqueness of our implementation lies in the fact that (i) the gate can be tuned for any phase shift ϕ from 0 to π and (ii) it operates optimally for all phase shifts. This combination of benefits was unprecedented and to the best of my knowledge even not yet repeated by any other laboratory.

28 16 CHAPTER. TUNABLE LINEAR-OPTICAL CONTROLLED-PHASE GATE

29 Chapter 3 Properties of the tunable controlled-phase gate In this chapter, we discuss the properties of the tunable c-phase gate. Once we have achieved its construction, we have investigated the behaviour of the gate in various situations and tested its capabilities. More specifically, we have focused on the reasons for its non-monotonic success probability and its entangling capability. Subsequent sections comment on two publications addressing the gate s properties. 3.1 Success probability of the controlled-phase gate Commenting on K. Lemr and A. Černoch, Optimal success probability of a tunable linear-optical controlled-phase gate, Phys. Rev. A 86, (01)[A1]. As already mentioned in the previous chapter, the success probability of the c-phase gate depends non-monotonically on the phase shift ϕ the gate is set to impose [47]. This result alone is rather counterintuitive since one would (naively) expect the success probability to decrease with the increasing phase shift since the larger the phase shift, the stronger is the gate s action. Indeed, the question of non-monotonic success probability was among the most frequent when I presented our results on various conferences. Although Konrad Kieling and his co-workers managed to identify the optimal success probability as function of the phase shift ϕ, their method was based on formal mathematical derivation and does not offer sufficient physical insight [47]. We have thus investigated this issue trying to find the physical explanations for the specific form of the success probability function. In 01, we published a paper showing that the gate s behaviour is a result of several simultaneous constraints one imposes on the gate [A1]. Firstly, one requires that the gate has uniform 17

30 18 CHAPTER 3. PROPERTIES OF THE TUNABLE CONTROLLED-PHASE GATE success probability independent on the input state mn P succ (ϕ)e iϕδ n1δ m1 mn, (3.1) where indexes m, n take logical values 0 or 1 and δ i j stands for the Kronecker delta. By following this condition, one assures that superpositions of eigenstates would not get deformed 1 1 α mn mn m=0 n=0 1 1 α mn Psucc (ϕ)e iϕδ n1δ m1 mn m=0 n=0 = P succ (ϕ) 1 m=0 n=0 1 α mn e iϕδ n1δ m1 mn, (3.) because the success probability can be factored out in front of the superposition term. This would not be the case if it were state dependent. In the experiment, it means that neutral density filters have to be applied to modes with higher average transmissivity to compensate for the less transmissive ones. Typically, the modes corresponding to logical states 0 that are not interacting in the gate have to be appropriately attenuated. Second requirement imposed on the gate comes directly from its definition (see Eq..1). The phase shift ϕ is applied only when both the control and target qubits are in the logical state 1. All other eigenstates shall not be mutually phase shifted. In experimental reality, the gate always imposes some intrinsic phase shift ξ to all its eigenstates mn e iξ mn mn. (3.3) This phase shift has to be equal for all eigenstates except for the 11 state, ξ 00 = ξ 01 = ξ 10. (3.4) The phase shift of the eigenstate 11 must take the form of ξ 11 = ξ 00 + ϕ. (3.5) The above mentioned conditions determine setting of filters and phase shifts in the Mach-Zehnder interferometers the gate consists of. The core of the gate is one particular Mach-Zehnder interferometer in which the photons bunch if both of them are in logical state 1 1. In case of other eigenstates, only one or none of the photons pass through this interferometer. Our analysis started by calculating the transformation amplitudes A x of the photons state in this interferometer if only one photon (denoted A 1 ) or both photons (denoted A ) travel through it. The first condition (state-independent success probability) results in the requirement that A = A 1. The phase difference between these amplitudes then determines the resulting phase shift of the gate ϕ. 1 referred to as the inner interferometer in Sec..3

31 3.1. SUCCESS PROBABILITY OF THE CONTROLLED-PHASE GATE 19 control signal { { BS S I C BS S C Figure 3.1: Conceptual scheme used to investigate the success probability of the tunable c-phase gate. The core interferometer mentioned in the text is formed by the beam splitters BS. Attenuations are denoted τ, τ S and τ C while phase shifts are labelled α I, α S and α C. Attenuation τ in one arm of the core interferometer together with a mutual phase shift between its arms α I are sufficient tools to tune the gate s phase ϕ while still satisfying the two constraints (see Fig. 3.1). We have identified a trade-off taking place when the gate phase shift ϕ is tuned. Let us start with the gate set for ϕ = π and imagine we wish to decrease the phase shift ϕ. In order to do that, we need to (i) change the phase difference α I between the arms in the core interferometer and (ii) insert filtering τ to one of these arms. Direct calculation reveals that with decreasing phase shift ϕ the value of α I approaches π/. As the result, the core interferometer yields lower transformation amplitudes A 1 and A since the closer the phase α I is to π/, the higher is the probability for the photons to leave the interferometer by incorrect output ports. In other words, destructive interference occurs in the correct output modes thus the gate s success probability decreases. At the same time however, the filtration τ has to be inserted which decreases the visibility of the core interferometer and thus increases the minimal transformation amplitudes A 1 and A obtained when the interferometer is set close to α I = π/. These two effects form the above mentioned trade-off with the minimum of the overall success probability appearing for ϕ π/3 (see Fig. 3.). By performing this analysis we were able to obtain identical success probability for a given phase shift ϕ as Kieling et al. did [44]. Our approach however allows to gain an insight into the gate s operation principle and explain why the success probability is non-monotonic. There are applications that do not require the c- phase gate to operate exactly according to the prescription (.1). For instance the requirement for uniform state independent success probability can be relaxed when the input state is known in advance [48]. In these cases the overall success probability of the gate can be increased.

32 0 CHAPTER 3. PROPERTIES OF THE TUNABLE CONTROLLED-PHASE GATE Figure 3.: The resulting success probability is a trade-off between the setting of the phase difference α I and filtering τ. The phase difference α I closes to π/ when the gate phase shift decreases from ϕ = π to lower values and thus the transformation amplitudes A 1 and A approach their respective minima (dotted line). At the same time, these minima increase because of inserting filtering τ that decreases the visibility of (destructive) interference inside the interferometer (dashed line).

33 3.. ENTANGLING EFFICIENCY OF LINEAR-OPTICAL QUANTUM GATES 1 3. Entangling efficiency of linear-optical quantum gates Commenting on K. Lemr, A. Černoch, J. Soubusta and M. Dušek, Entangling efficiency of linear-optical quantum gates, Phys. Rev. A 86, 0331 (01) [A]. It has being already explained in the preceding chapter, that the c-phase gate has the entangling capability allowing it to entangle an originally separable state of the target and control qubits [39]. While this property was already well known, until our optimal tunable c-phase gate was constructed, it was impossible to study it with respect to the phase shift ϕ. In the paper commented in this section, we have analysed the output state entanglement depending on the phase shift ϕ as well as in relation to the corresponding success probability [A]. We have studied the following scenario: an initially separable state of the target and control qubits is inserted into the c-phase gate parametrised by its phase shift ϕ. Once the gate processes this state, we calculate the amount of entanglement contained in the output state. Various measures of entanglement can be used for this purpose. We have selected the negativity N as a suitable candidate namely because it is easily calculated from the density matrix using the formula N ( ˆρ) = ˆρT A 1, (3.6) where denotes the trace norm and T A stands for partial transposition of the density matrix [49]. Moreover, negativity is also known to be convex N p m ˆρ m p m N ( ρˆ m ), (3.7) m which also means that a mixture of pure entangled states can not have negativity larger then any of the pure states it consists of. The output state negativity is not only a function of the gate parameters, but also of the specific input state. When solely the gate s action is to be analysed, it is necessary to eliminate the influence of the input state choice. It is then customary to use the so-called entangling power as a measure of quantum gate s capability to entangle input states [50 5]. This measure is simply defined as the supremum of achievable entanglement of output states taken over all possible separable input states m E p = sup {N ( ˆρ OU T )}. (3.8) ˆρ IN S In this equation, S denotes the set of separable input states ˆρ IN while ˆρ OU T denotes the output state density matrix obtained from ˆρ IN by the gate s transformation G ˆρ OU T = G ( ˆρ IN ). (3.9) The problem is that the entangling power does not take into account the probabilistic nature of linear-optical quantum gates. When comparing two probabilistic

34 CHAPTER 3. PROPERTIES OF THE TUNABLE CONTROLLED-PHASE GATE p s (left axis) E p (left axis) E eff (right axis) /4 / 3 /4 Gate phase shift 0.00 Figure 3.3: Entangling power E p, entangling efficiency E eff and success probability p s of a tunable c-phase gate as functions of its phase shift ϕ. Experimental data are depicted using markers while lines visualise theoretical values. gates, the one with higher entangling power can have significantly lower success rate resulting in fact in lower entanglement yield average amount (e.g. negativity) of entanglement generated per a given number of trials (e.g. input state insertions). Let us imagine two gates: the first one has higher entangling power, but the second has higher success rate. The question is now apparent, which one is better in generating entangled states? To answer this question, we have generalised the entangling power to take into account the probabilistic nature of the gates. We have proposed a new measure the entangling efficiency defined as E eff = sup {p s ( ˆρ IN ) N ( ˆρ OU T )}, (3.10) ˆρ IN S where p s ( ˆρ IN ) denotes the success probability. Since the c-phase gate in its standard form has equal success probability for any given input state, the success probability can be factored out in Eq. (3.10) and the entangling efficiency becomes just a product of entangling power and success probability. We have shown that both the entangling efficiency and entangling power are monotonic functions of the phase shift ϕ with maxima at ϕ = π. This result alone is not particularly surprising, but allows to confront experimentally measured data using our tunable c-phase gate with theoretical predictions (see Fig. 3.3). A more interesting situation can be observed when we generalise the gate by dropping the requirement on state independent success probability. As explained in Sec. 3.1, the success rate can be increased for some input states by removing the ballast filters from more transmissive modes. As a result, the success probability

35 3.. ENTANGLING EFFICIENCY OF LINEAR-OPTICAL QUANTUM GATES max {N P s } max {N} P s N N 0 /1 /6 /4 /3 5 /1 / Angle s P s Figure 3.4: Success probability P s (full line), negativity N (dotted line), and their mutual product (dashed line) are plotted against the parameter of input state θ s for ϕ = π and θ c = π/. All the functions are symmetric in the sense that the same results would be obtained when exchanging θ s and θ s. of the transformation becomes state-dependent 00 00, 01 4 p s (ϕ) 01, 10 4 p s (ϕ) 10, 11 p s (ϕ)e iϕ 11. (3.11) At this point one can no longer factor out the success probability from the maximisation in Eq. (3.10). Maximum of output state negativity alone (therefore entangling power) can be obtained for a different separable input state then the maximum of the product of output state negativity and success probability (entangling efficiency). We have searched for the particular states maximising both the equations for entangling power (3.7) as well as for entangling efficiency (3.10). Input qubit states were parametrised by angles θ s,c and ϑ s,c ψ s,c = cos θ s,c 0 s,c + e iϑ s,c sin θ s,c 1 s,c, (3.1) where indexes s and c stand for the signal and control qubits respectively. A straightforward calculation reveals that maximum of negativity, thus the entangling power, is obtained for different angles then the maximum of the product of negativity and success probability (entangling efficiency). In order to visualise this effect, we have fixed parameters of the control qubit θ c = π/4, ϑ c = 0 and depicted negativity, success probability and their product as functions of the signal qubit parameter θ s (see Fig. 3.4). Note that this result is valid for all values of the gate s phase shifts ϕ. In the paper, we thus argue that for practical applications, where highest entanglement yield is to be obtained using probabilistic quantum gates, the entangling efficiency is a more suitable measure. It gives a more representative description of the gate s entangling capabilities.

36 4 CHAPTER 3. PROPERTIES OF THE TUNABLE CONTROLLED-PHASE GATE

37 Chapter 4 Applications of the tunable controlled-phase gate It has been already stated in preceding chapters, that the controlled-phase gate takes a prominent place in the toolbox for universal quantum circuit design. Further to that, it also plays a crucial role in other quantum information related tasks. In 1996, Charles Bennett and his colleagues proposed to use the CNOT gate for entanglement purification [53], an important resource for quantum repeaters [54]. Soon afterwards, David Deutsch et al. invented a strategy for privacy amplification of quantum cryptography over noisy channels where the CNOT gate also plays a key role [55]. The entangling capability of the c-phase gate can be used in quantum state engineering, as it has been shown by Franson et al. [56] or in quantum non-demolition measurement as Pryde et al. showed experimentally [48]. L. Roa et al. proposed a quantum teleportation scheme based on the CNOT operation [57]. Recently, C. Vitelli and colleagues have experimentally demonstrated a protocol for quantum state fusion using a sequence of three CNOT gates 1 [58]. The above mentioned examples demonstrate, that the c-phase gate with its numerous applications is an essential tool for quantum information processing. The tunability of the phase shift ϕ gives this gate even a more versatile nature. It is thus reasonable to expect that replacing the non-tunable c-phase gate with its more powerful tunable version should open ways for new applications or at least for improvement of the already known ones. In 013, the applicant was awarded a post-doctoral grant by the Czech Science Foundation specifically dedicated to search for further applications of the tunable c-phase gate. As a result he has published a number of papers addressing this topic. Five of these papers are commented in detail in this chapter. 1 Quantum state fusion is a protocol transcoding two qubits on originally two different carriers (e.g. photons) into one single carrier supporting a four-dimensional Hilbert space (e.g. combination of polarisation and dual-rail encoding on a single photon). 5

38 6 CHAPTER 4. APPLICATIONS OF THE TUNABLE CONTROLLED-PHASE GATE 4.1 Preparation of Knill-Laflamme-Milburn states Commenting on K. Lemr, Preparation of Knill-Laflamme-Milburn states using a tunable controlled-phase gate, J. Phys. B 44, (011) [A3]. One notable property of the controlled-phase gate is its entangling capability [39]. The gate can create an entangled quantum state of the control and target qubits from an initially separable state. This property is particularly useful for quantum state engineering, where one s goal is to generate specific, possibly complex, quantum states from originally less complex ones. In their seminal paper from 001, Knill, Laflamme and Milburn proposed a method that significantly increases success probability of linear-optical quantum computing [37]. Their proposal is based on using a particular complex entangled quantum state as an ancilla. This state, known as the Knill-Laflamme-Milburn (KLM) state, can be expressed in terms of logical qubits in the form of ψ KLM = n α j 1 j 0 n j, (4.1) j=0 where n is the number of qubits involved. In the original KLM state definition, the superposition of individual terms was balanced with α j = 1 n+1 for j = 0...n. It was however shown later by Franson et al. that non-equal values of α j provide further benefits [59]. Although Knill, Laflamme and Milburn demonstrated usefulness of this quantum state, they did not give any specific recipe for its preparation. It was until several years later that Franson et al. theoretically proposed how to generate these states using quantum c-phase gates [56]. Their method is however not optimal since at that time, they could not yet benefit from the optimisation based on suitable tuning the phase shift ϕ. A conceptually different method for generation of two-qubit KLM states uses a beam splitter transformation of a partially entangled two-photon states [A1, A13]. This method can not however by simply generalised to generate higher n-qubit KLM states. In my paper from 011, I have investigated the possibility to use a tunable controlled-phase gate for generation of the KLM states [A3]. Although this paper was published during the last year of my Ph.D. studies, it did not take part in my Ph.D. thesis. The non-monotonic dependence of the success probability of the gate as a function of its phase shift is reflected in the scheme optimisation. This way the overall success probability is maximised within the framework of linear optics. In the first part of my paper, I start by deriving a specific recipe allowing to generate two-qubit KLM states ψ KLM = α α α 11, (4.)

39 4.1. PREPARATION OF KNILL-LAFLAMME-MILBURN STATES 7 (a) (b) Figure 4.1: (a) Conceptual scheme of the protocol for generation of two-qubit KLM states using the tunable c-phase gate. (b) Generalisation of the scheme allowing to incorporate an extra qubit into an n-qubit KLM state thus generating an (n + 1)-qubit KLM state. where α j (for j = 0, 1, ) are arbitrary complex amplitudes following the normalisation condition j=0 α j = 1. The idea is to first prepare two qubits ψ s and ψ c in separable states of the form of ψ s,c = cos θ s,c 0 s,c + e iφ s,c sin θ s,c 1 s,c (4.3) and then subject these qubits to a tunable c-phase gate (see Fig. 4.1a). In the paper, I give explicit formula relating parameters of these qubits (θ c,s, φ c,s ) and the gate phase shift ϕ with the resulting parameters of a two-qubit KLM state (α j for j = 0, 1, ). It can be shown that the value of the phase shift ϕ limits the maximally attainable ratio α 0 /α 1. The protocol thus searches for such phase shift ϕ that among all phase shifts allowing to generate the required KLM state can be implemented with highest success rate. Subsequently, in the next section of the paper, I also present a way of how to generalise the scheme to prepare KLM states of higher number of qubits. This generalised scheme works iteratively. It adds one new qubit into an n-qubit KLM state thus yielding an (n + 1)-qubit KLM state (see Fig. 4.1b). The paper investigates some optimisation options of this generalised scheme. Given the parameters of target KLM state, the phase shift of each c-phase gate and parameters of various single-qubit gates are set to provide maximum efficiency. In comparison to the work by Franson et al. [56], my scheme increases the overall success probability of two-qubit KLM state preparation by tens of percents, depending on the specific KLM state. This efficiency improvement is even more pronounced in the case of KLM states composed of more qubits. Other scientists have subsequently continued in this research direction. In 01, Liu-Yong Cheng et al. adapted this scheme for preparation of KLM states on the platform of light interacting with atoms inside a cavity [60]. In 014, Qi-Gong Liu et al. showed how to generate the KLM states with superconducting qutrits [61].

40 8 CHAPTER 4. APPLICATIONS OF THE TUNABLE CONTROLLED-PHASE GATE signal input Ψ s control input quantum router Ψ c = cos θ 0 + e ıϑ sin θ 1 signal output 1 A } 1(θ, ϑ) Ψ s 1 +A (θ, ϑ) Ψ s Figure 4.: Conceptual scheme of a quantum router. Signal qubit is routed into superposition of two output modes depending on the state of the control qubit. 4. Linear-optical quantum routers Commenting on K. Lemr and A. Černoch, Linear-optical programmable quantum router, Opt. Comm 300, 8 85 (013) [A4] and K. Lemr, K. Bartkiewicz, A. Černoch and J. Soubusta, Resource-efficient linear-optical quantum router, Phys. Rev. A 87, (013) [A5]. Light is a particularly efficient information carrier. It is thus hardly surprising that modern classical communications networks routinely rely on light or other invisible electromagnetic radiation propagating either in fibres or in free space [6]. Similarly in quantum communications, light plays a prominent role [63, 64]. So far however the quantum communications networks were limited, at least dominantly, to a simple point-to-point geometry. If these networks are to replace their classical analogues one day, we need to develop techniques that are necessary for their complex operations and scalability. In classical communications networks, the information is steered from the sending to receiving party using active elements such as routers or switches. Based on some control information (e.g. the IP address), these active elements decide to which output port the information shall be directed [65]. We have investigated the quantum analogue to a classical router the quantum router. A conceptual scheme of a quantum router is depicted in Fig. 4.. In contrast to its classical counterpart, the quantum router admits qubits as both the signal and control information. Further to that, the quantum router has to benefit from the principle of superposition. Therefore it has to be possible to route the message coherently into multiple output ports. Let us denote the state of a signal qubit ψ s and the state of a control qubit ψ c = A 1 0 +A 1, where A 1 + A = 1. The quantum router implements the transformation ψ s IN ψ c IN A 1 ψ s OUT1 + A ψ s OUT, (4.4) where indexes IN, OUT1 and OUT denote input, first output and second output ports respectively. Researchers have already investigated the possibility to implement quantum routers using light-matter interaction [66 68]. Such interaction is however ex-

41 4.. LINEAR-OPTICAL QUANTUM ROUTERS 9 PPG Φ c λ/@.5 PBS 1 A 1 Ψ s 1 λ/@0 A Ψ s Ψ s 1 PBS 1 PPG Φ c1 Figure 4.3: Linear optical scheme of a quantum router using two identical control qubits ψ c to route one signal qubit ψ s. PPG stands for the programmable phase gate as explained in the text. perimentally very challenging. There exist also some proposals for all-optical quantum routers, but in these cases the control information is either classical, not quantum [69] or the signal information collapses depending on the control state measurement [70]. In our first paper on this topic, we have proposed a scheme for a genuine quantum router performing the transformation as prescribed in Eq. (4.4) [A4]. This scheme however requires two identical control qubits to route one signal qubit (see Fig. 4.3). First, the signal qubit is split into two spatial modes which then interact with the control qubits inside two programmable-phase gates. These gates imprint the amplitude ratio A 1 /A of the control qubit state ψ c into the spatial degree of freedom of the signal qubit allowing it to leave the router coherently by two output ports. The entire routing operation can be implemented with success probability of 1/4. We were however dissatisfied with the fact that this scheme requires two control qubits for its operation. To address this issue, we have devised yet another scheme published in our second paper on quantum routing [A5]. We started this second paper by summarising a set of five requirements that a quantum router shall fulfil: Both the signal and control information have to be stored in quantum objects (qubits), therefore routers using classical information to route quantum signal are considered only semi-quantum routers. The signal information is unchanged under the routing operation, the degree of freedom used to store the signal qubit information has to be kept undisturbed. The router has to be able to route the signal into a coherent superposition of both output modes.

42 30 CHAPTER 4. APPLICATIONS OF THE TUNABLE CONTROLLED-PHASE GATE PPG λ/ s 1 A 1 Ψ s 1 signal outputs A Ψ s s NDF T = 1 λ/ PBS λ/ Ψ s s IN signal input PBS 1 1 generalized c-phase gate: PDBS QND generalized c-phase gate control input PBS c PPG: D/A Φ c c QND: PBS (4x) a 1 D/A all λ/ λ/ λ/ λ/ @45 D/A a Figure 4.4: Scheme of our second linear-optical quantum router. Both the control and signal qubits are encoded into polarisation states of individual photons. By means of their interaction in a tunable controlled-phase gate and a programmable phase gate, the signal qubit is directed into a superposition of two spatial modes while preserving its original polarisation-encoded signal state. The router has to work without any need for post-selection on the signal output. If the router is probabilistic, successful operation can be identified by detection on the control state. To optimise the resources of the quantum network, only individual control qubit is required to direct one signal qubit. (direct quote from Ref. [A5]) All theoretical and experimental proposals for quantum routers known at that time only fulfilled these criteria partially. We have thus decided to propose a linear-optical quantum router which meets all the above mentioned requirements and requires only one control qubit per one signal qubit. In our second paper on this topic, we give an explicit recipe for construction of such device using linear optics and discrete photons. The core element of our proposal is the c-phase gate in which the interaction between the signal and control information takes place. Our scheme benefits from the fact that we can use the c-phase gate in its generalised form with state-dependent success probability. This way we are able to increase its success rate from 1/9 to 1/6. The success

43 4.3. APPLICATIONS OF QUANTUM ROUTERS 31 probability of the entire routing procedure is then 1/4. We subsequently show that at the expense of some routing limitations, the c-phase gate can be tuned to values of phase shift ϕ smaller then π and thus one can increase the overall success probability of the scheme even more. We have also analysed the possibility to generate complex entangled quantum states using a repetitive action of this quantum router. Independently, a quantum optics group from Rome has performed an experimental implementation of a formally identical operation quantum state fusion [58]. Their scheme succeeds with probability of 1/8 if a feed-forward operation is implemented on the output state. Although their scheme is also capable of performing quantum routing operation, the authors have not mentioned that in their paper and have focused solely on quantum state fusion. Meanwhile, our definition of a fully functional quantum router and the set of five criteria have been adopted by other researchers [71]. Subsequently, in 014, Jing Lu et al. proposed a single-photon quantum router based on the interaction between photons and an atom inside a cavity [7]. Wei- Bin Yan and Heng Fan then generalised this concept to an arbitrary number of output ports [71], while Xiang Zhan and colleagues have pointed out to some similarities between the operation of quantum routing and quantum walk [73]. Ye-Wang Chen and Qing Lin then suggested to use a non-linear optical interaction to implement the quantum router [74]. Last year, X. X. Yuan et al. experimentally implemented our scheme and tested its functionality on a limited set of signal and control qubit states [75]. 4.3 Applications of quantum routers Commenting on K. Bartkiewicz, A. Černoch and K. Lemr, Using quantum routers to implement quantum message authentication and Bell-state manipulation, Phys. Rev. A 90, 0335 (014) [A6]. While the original function of a quantum router is, as its name suggests, the information routing in complex quantum communications networks, it has a much broader range of applications. In the paper discussed in this section, we have analysed two of these applications: quantum message authentication and Bell state non-demolition manipulation [A6]. For the above mentioned tasks, we first needed to define an inverse device to the quantum router. We have named this device the quantum decoupler and it allows to decouple two qubits encoded into different degrees of freedom of one physical object. On the platform of linear optics, the proposed decoupler splits polarisation and spatial mode encoded qubits stored within one single photon. Conceptual scheme of the quantum decoupler is shown in Fig First application of the quantum router we have analysed was a protocol for quantum message authentication. In classical communications, one can use dig-

44 3 CHAPTER 4. APPLICATIONS OF THE TUNABLE CONTROLLED-PHASE GATE Figure 4.5: Conceptual scheme of a quantum decoupler for photons. Two qubits are stored in the state of the signal photon: one is encoded in its spatial degree of freedom and the second in polarisation. Quantum decoupler facilitates interaction between this signal input state and an ancillary photon leading to transcription of the spatial mode encoded qubit to the polarisation state of the ancillary photon while the polarisation signal qubit remains stored in the signal photon. ital signatures to authenticate the message sending party [76]. In this case the message and corresponding signature are usually readable and verifiable without any secret shared between the sending and receiving party. It has been shown that quantum message signing is possible in arbitrated schemes [77], but it is impossible to achieve it in a non-arbitrated way [78]. We have focused on a very closely related task of quantum message authentication [79]. In this case a secret shared between the sending and receiving party allows to verify the authenticity of a quantum message. As prescribed in our protocol, the sending party prepares two strings of quantum bits. One string contains the message while the other contains a random authentication key. Then the sending party inserts first qubits from both these strings into the quantum router. Depending on a secret classical key shared with the receiving party, either the message or the authentication qubit is inserted as the control or target qubit into the router. The output of this quantum router is connected via a quantum communications network to a quantum decoupler of the receiving party. In this decoupler, the message and authentication qubits are split and a measurement is performed on the authentication qubit to verify that the message is genuine. Note that the shared secret between the sending and receiving parties can be established using standard quantum cryptography protocols [80 8]. In the paper we present a detailed analysis of possible attack on this protocol. We show that the probability of successfully falsifying a quantum

45 4.3. APPLICATIONS OF QUANTUM ROUTERS 33 1 Figure 4.6: Linear optical implementation of a Bell state splitter attached to the output of a quantum router. A two-qubit state is first subjected to a quantum router yielding one output photon with both polarisation and spatial-mode encoded qubits. Detection of this photon on any particular detector projects the original two-photon state on one specific Bell state. message P C transmitted in this way decreases exponentially with the length of the message as well as with the number of randomly added decoy message qubits P C = 3 n 3 d, (4.5) 4 8 where n stands for the number of message qubits and d for the number of decoy states. We realise that this protocol is feasible even without a quantum router, but we also stress out the benefits of using the quantum router especially with highly lossy quantum communications channels. The second investigated application of a quantum router is the task of nondemolition Bell state manipulation [83]. Bell states are four maximally entangled two-qubit states of the form of Φ ± 1 ( 00 ± 11 ), (4.6a) Ψ ± 1 ( 01 ± 10 ). (4.6b) They form a complete basis in the Hilbert space of two-qubit states. These states are often used in various quantum information processing tasks, namely in quantum teleportation [84, 85]. Given their wide range of applications, it is useful to investigate the procedure of their complete discrimination. That is to project an unknown two-qubit state on one of the four Bell states [83, 86]. This task can not be achieved deterministically with linear optics if only one degree of freedom of a photon pair is used to encode the Bell state [87]. In our paper we show how Note that if more degrees of freedom are used simultaneously, Bell state discrimination can be implemented deterministically even with linear optics alone [88].

46 34 CHAPTER 4. APPLICATIONS OF THE TUNABLE CONTROLLED-PHASE GATE Figure 4.7: Bell-state manipulation using a quantum router and decoupler. A two-qubit state is first processed by the quantum router. Subsequently, the output signal photon is subjected to a Bell state splitter (see Fig. 4.6). Set of four neutral density filters and phase shifters is used to adjust amplitude and phase separately in each individual mode corresponding to one Bell state. The signal is recombined on an inverse Bell-state splitter and finally two separate photons are recreated using the quantum decoupler. to achieve it at least probabilistically using a quantum router [A6]. The router allows to re-encode the Bell state of two photons into one single photon using two polarisation and two spatial modes. A simple linear-optical scheme can then be used to direct this photon onto one of four detectors (see the scheme for a Bell state splitter in Fig. 4.6). Detection of the photon on any of these detectors corresponds to the projection of the original two-photon state onto one of the Bell states. We further develop the idea of the Bell state discrimination and propose a scheme for non-demolition Bell state manipulation. The goal is to perform an operation of the form of Φ τ 1 e iϕ 1 Φ, (4.7a) Φ + τ e iϕ Φ +, (4.7b) Ψ τ 3 e iϕ 3 Ψ, (4.7c) Ψ + τ 4 e iϕ 4 Ψ +, (4.7d) where τ m and ϕ m for m = 1,, 3, 4 are independent amplitude transmittances and phase shifts respectively. Such operation in its general form can not be implemented by local operations directly on the two entangled photons. One can easily see that local phase shifts and amplitude damping on the photons affect more that one single Bell state. For instance the attempt to completely block the state Φ results in extinguishing the Φ + state as well. To circumvent this issue, we propose to subject the photons to a quantum router yielding one photon encoded both in polarisation and spatial mode. Similarly to the discrimination procedure, this photon is split into four paths depending on the initial Bell state. Phase shifts and amplitude damping is then performed independently in these

47 4.4. IMPLEMENTATION OF CONTROLLED-UNITARY GATES 35 four modes. Finally, the inverse to this splitting is performed and the state of the photon is back re-encoded into two individual photons using a quantum decoupler (see conceptual diagram in Fig. 4.7). 4.4 Implementation of controlled-unitary gates Commenting on K. Lemr, K. Bartkiewicz, A. Černoch, M. Dušek and J. Soubusta, Experimental Implementation of Optimal Linear-Optical Controlled-Unitary Gates, Phys. Rev. Lett. 114, (015) [A7]. One significant setback of quantum computing with linear optics is its inherently probabilistic nature [1]. It has been shown that a number of important quantum gates can only be implemented with success probability lower than one. This property limits severely the scalability of quantum gates on this platform. Various techniques addressing this issue have been investigated [89], for instance the Knill-Laflamme-Milburn approach discussed in Sec 4.1 [37]. Processing of quantum information inside a quantum circuit can be regarded as a general unitary transformation of signal qubits conditioned on the state of controlled qubits controlled-unitary transformation. It has been established that any such transformation can be achieved using only the controlled-sign (or CNOT) gate and unconditional single-qubit rotations [38]. Such decomposition is not necessarily the only one possible. On probabilistic platforms, such as linear optics, it is of vital importance to search for decompositions that maximise the success probability. In our research paper [A7], we have considered one particular example of these conditional unitary transformations and derived an optimal decomposition using the tunable c-phase gate. We have focused on unitary transformations of a signal qubit ψ s conditioned on the state of a control qubit. Such operations on the signal and control qubits are expressed as ψ s 0 ψ s 0, (4.8a) ψ s 1 Ŵ ψ s 1, (4.8b) where Ŵ represent the signal qubit unitary transformation. The operator Ŵ is parametrised by three real angles δ, β and γ and in basis spanned by logical states 0 and 1 it takes the matrix form of e i(δ+β) cos γ e i(δ β) sin γ Ŵ = e i( δ+β) sin γ e i( δ β). (4.9) cos γ In 1995, Barenco et al. proposed a general circuit implementing the controlledunitary transformation using two c-phase gates and several single-qubit rotations [38]. Note that at that time, only c-phase gates with fixed phase shift ϕ = π were known. On the platform of linear optics, the optimal success probability of

48 36 CHAPTER 4. APPLICATIONS OF THE TUNABLE CONTROLLED-PHASE GATE = Figure 4.8: Conceptual scheme for implementation of a controlled-unitary transformation Ŵ using the tunable c-phase gate Ẑ(ϕ) and unconditional rotations ˆR Z (α) and ˆR Y (θ). the c-phase gate set to ϕ = π is 1/9. Using this gate twice means that the overall success probability ends up to be 1/81. Probabilistic nature of these c-phase gates also requires some non-demolition detection to be implemented in between to post-select on correct outputs of the first c-phase gate [85, 90, 91, A14]. This non-demolition detection is achieved typically with success probability of 1/ rendering the overall success rate of the scheme 1/16. With the capability to tune the c-phase gate we have obtained an additional degree of freedom. Our idea was to use this degree of freedom to increase efficiency of general controlled-unitary transformations. Although some theoretical proposals were already known at that time, they did not provide any explicit recipe for construction of a general controlled-unitary gate using the tunable c-phase gate [9]. Simultaneously, another approach was tested experimentally by Lanyon and his colleagues, but with lower then optimal success probability [46]. In 015, we derived an optimal scheme for implementation of linear-optical controlled-unitary transformation [A7] (see Fig. 4.8). We have also tested our scheme experimentally using qubits encoded into polarisation states of individual photons. The success probability of our scheme varies depending on the concrete controlled-unitary transformation. In average we reach success rate of about 14%. In contrast to Barenco et al. [38], this represents an significant improvement, more than one order of magnitude. The idea behind our approach is straightforward. The signal qubit is first transformed by two unconditional rotations. In the basis of logical states, these two rotations take the matrix forms 1 0 ˆR Z (α) = 0 e iα (4.10) and cos θ ˆR Y (θ) = sin θ sin θ cos θ (4.11) respectively. In the next step, the signal qubit together with the control qubit are subjected to a c-phase gate. Finally, reverse rotations ˆR Y ( θ) and ˆR Z ( α) are imposed on the signal qubit. As a result, if the control qubit reads 0, the c-phase gate does not change the signal state and the rotations in front and behind it thus cancel each other. On the other hand, if the control qubit reads 1, the combined

49 4.4. IMPLEMENTATION OF CONTROLLED-UNITARY GATES 37 SIGNAL IN CONTROL IN MT1 HWP PBS1 HWP QWP BD piezo BD BDA F HWP HWP MT BDA1 piezo F1 HWP HWP PBS D1V DV PBS HWP QWP PBS D1H DH SIGNAL OUT CONTROL OUT Figure 4.9: Schematic drawing of the experimental setup for a controlled-unitary gate. The components are labelled as follows: MT motorised translation, HWP half-wave plate, QWP quarter-wave plate, PBS polarising beam splitter, BDA beam divider assembly, BD beam divider, F neutral density filter, D detector. For more details, see Sec..3. effect on the signal qubit is a general unitary transformation. In the paper, we give explicit relations between the parameters (δ, β, γ) as used in Eq. (4.9) and (α, θ, ϕ). We have tested our model experimentally using a modified setup for the c- phase gate (see Fig. 4.9). Photons were generated in the process of spontaneous parametric down-conversion and then coupled into the setup using single-mode optical fibres. State preparation and single-qubit unconditional rotations are implemented using sets of half and quarter-wave plates. The c-phase gate is implemented as described in detail in Sec..3. We have tested various conditional unitary transformations for various phase shifts ϕ of the c-phase gate. To fully characterise the implemented transformation, we have performed a complete process tomography of the signal mode and estimated corresponding process matrices [93 95]. An example of such process matrices for both the control qubit in the logical state 0 and 1 is depicted in Fig Typical fidelity of these transformations reached about 90% which proves a good agreement between theory and experiment. To demonstrate the scalability of our approach, we have presented an explicit way of how to construct a controlled-unitary transformation with more then one control qubit. In these cases the tunable c-phase gate still provides a measurable improvement over all previously known schemes.

50 38 CHAPTER 4. APPLICATIONS OF THE TUNABLE CONTROLLED-PHASE GATE HH HV VH VV HH (a) HV VH VV 0 HH HV VH VV HH (b) HV VH VV HH HV VH VV HH (c) HV VH VV 0 HH HV VH VV HH (d) HV VH VV Figure 4.10: Estimated process matrices for ϕ = 3π 4, θ = π and α = 0 (a) with control qubit 0 and (b) with control qubit 1. Their respective theoretical predictions are presented as (c) and (d). Moduli of matrix elements are visualised by bar heights and their phases by arrow directions.

51 Chapter 5 Programmable controlled-phase gate Commenting on K. Lemr, K. Bartkiewicz, A. Černoch, Scheme for a linear-optical controlled-phase gate with programmable phase shift, J. Opt. 17, 150 (015) [A8]. In all the previous sections, we have assumed that the gate s parameters are hard-coded into the optical setup itself. The target phase shift ϕ is thus achieved by specific setting of individual linear-optical components. Hence classical information is used to parametrise the gate. In our recent paper, we have proposed a linear-optical setup where the phase shift ϕ is encoded into a bit of quantum information which is then used to program the gate (see conceptual scheme in Fig. 5.1). Quantum gates that have their operation parametrised by quantum information are referred to as programmable gates [96]. Parameters of imposed transformations are delivered in the form of one or several so-called program qubits. Theoretical framework of programmable gates has been addressed in several pa- Figure 5.1: Conceptual scheme of a programmable c-phase gate. T, C and P denote target, control, and program ports, respectively. The phase shift ϕ is encoded into the state of the program qubit as described in Eq. (5.1). 39

52 40 CHAPTER 5. PROGRAMMABLE CONTROLLED-PHASE GATE pers that revealed a number of interesting benefits of this concept [96 98]. Since quantum information can be teleported, one can also teleport gate s parameters stored in program qubits. This can be regarded as the possibility to teleport a specific transformation (or gate) from one place to another. In a sense this is similar to software distribution on current classical computers. Moreover note that the phase shift value is a real number and thus it takes infinite amount of classical information to transmit it with unlimited precision. Using quantum information, one qubit suffices to achieve this task. In a recent experiment, Mičuda et al. have demonstrated a programmable single-qubit gate [99]. In their experiment, the target qubit state is transformed (rotated) depending on the state of the program qubit. Note that in such programmable single-qubit gate, the programmed transformation is imposed on the signal state without being conditioned on any control qubit. This is an important conceptual difference between the programmable single-qubit gate on one hand and the c-phase gate on the other. As it has been emphasised in Chapter, the c-phase gate belongs to the set of universal gates [38]. Together with single-qubit rotations it forms a sufficient set of tools to design any quantum computing circuit. If the circuit is to be versatile, all its parameters shall be programmable by a register of program qubits. Both theoretical and experimental study of programmable single-qubit rotations has been addressed in Refs. [99, 100]. It remains to investigate, how to make the c-phase gate programmable as well. In the commented paper, we have proposed a linear-optical scheme for a programmable c-phase gate (see Fig. 5.) [A8]. It is inspired by both the Mičuda et al. experiment [99] and a non-optimal and non-programmable c-phase gate constructed by Lanyon et al. [46]. The scheme operates with qubits encoded into polarisation states of individual photons. Besides the target and control qubits, an additional program qubit in the state Ψ p = 1 0 e iϕ 1 (5.1) is used. The phase shift between logical states 0 and 1 of the program qubit then directly translates into the setting the gate s phase shift ϕ. The principle of operation of our programmable c-phase gate can be explained as follows: if in logical state 1, the target qubit enters an auxiliary spatial mode, where it interacts with the control qubit in a CNOT gate. This CNOT gate is implemented by a partially polarising beam splitter PPBS [44] enveloped by two Hadamard gates. If the control qubit is in the logical state 1 it makes the target qubit swap from 1 to 0. Subsequently the target qubit undergoes a programmable phase gate where the interaction with the program qubit leads to the target state transformation 0 e iϕ 0, (5.) 1 1.

53 41 Figure 5.: Scheme for a linear-optical programmable c-phase gate. The target, control and program qubit enter the setup by T IN, C IN and P IN respectively. The target and control output ports are denoted T OUT and C OUT. The program qubit is projected by detector D onto diagonal/anti-diagonal polarisation. Polarising beam splitters PBS x (x = 1,, 3) transmit horizontally polarised photons while reflecting vertical polarisation. The partially polarising beam splitter PPBS has unit transmissivity for horizontal polarisation and amplitude transmissivity t V = 1/ 3 for vertical polarisation. Depending on the program qubit detection outcome, a feed-forward in the form of a phase shift is either imposed by the phase light modulator PLM or not. So only if previously swapped in the CNOT gate (because the control qubit is in the state 1 ), the target qubit acquires the phase shift ϕ. Finally, the target auxiliary mode gets recombined with its original spatial mode. Note that the gate succeeds when the target and control photons leave by designated output ports and the program qubit is detected and measured. In this case the entire scheme implements the usual c-phase transformation as defined in Eq. (.1). Without any feed-forward and with one fixed signal output port, our scheme works with success probability of 1/48. This can however be improved by two independent steps. First step involves a feed-forward transformation applied to the output target state depending on the measurement outcome on the program photon. Second step involves allowing the target photon to leave the setup by either of the two output ports of the final polarising beam splitter PBS. Each optimisation step increases the overall success probability twice and thus one can reach the success probability of 1/1 if both optimisation steps are used. This success probability is constant for any phase shift ϕ of the gate.

54 4 CHAPTER 5. PROGRAMMABLE CONTROLLED-PHASE GATE

55 Chapter 6 Linear-optical qubit amplifiers In previous chapters, we have already discussed the suitability of discrete photons for construction of quantum communications networks. We have promoted the role of the c-phase gate in these networks and given numerous examples of its applications. This chapter presents a closely related topic of quantum amplifiers without which quantum communications can not be achieved over longer distances. Further to that, quantum amplifiers allow to post-select correct outcomes of probabilistic quantum gates without disturbing the output qubits [91, A14]. This is particularly useful if the c-phase gate is to be used repetitively in a quantum circuit. Contemporary communications channels are burdened by various sorts of losses and sources of noise [101]. Efficient transmission of information across large distances thus necessitates the use of repeaters and amplifiers. These devices usually measure the transmitted signal and broadcast its copy on a stronger carrier wave. Quantum communications channels are also facing similar technological issues that need to be solved by quantum repeaters and quantum amplifiers. In contrast to classical information, qubits can not be simply copied and retransmitted. A fundamental theorem in quantum physics known as the no-cloning theorem forbids exact duplication of an unknown quantum state [19]. The amplification of quantum information thus requires a completely different approach. There are two conceptually distinct strategies that quantum amplifiers adopt. The first strategy involves approximate duplication of the signal quantum state [10]. Devices that create these approximate copies are called quantum cloners [103]. The fact that the copies of the amplified state are not perfect means that the amplifier inevitably adds some amount of noise. In principle, such amplification procedure can be implemented deterministically. In one of our experiments, we have experimentally demonstrated this strategy on discrete photons using linear optics [A15]. On the platform of linear optics, the cloning operation is however probabilistic and thus in order to actually increase the photon rate, we had to combine it with randomly added mixed qubit states. The second amplification strategy is based on filtering the signal quantum 43

56 44 CHAPTER 6. LINEAR-OPTICAL QUBIT AMPLIFIERS state [104]. More specifically, let us consider a qubit of quantum information in a general state ˆρ. By propagating over a lossy channel, this qubit gets mixed with a vacuum component 0 0 ˆρ p 1 ˆρ + p (6.1) The goal of a filtration-based amplification is to reduce the contribution of this vacuum component p 1 ˆρ + p p 1G N ˆρ + p 0 0 0, (6.) N where G denotes the amplification gain and N is a normalisation factor. Perfect noiseless amplifiers fulfil the condition ˆρ = ˆρ. Amplification operation described by (6.) is inevitably probabilistic and its success has to be heralded for instance in the form of a particular measurement outcome in an ancillary mode. Devices working on this principle are thus often referred to as heralded quantum amplifiers [105]. Heralded qubit amplifiers do not actually increase the signal energy. They are nonetheless indispensable for instance in device-independent quantum key distribution [106], where they allow to close the detection loophole [107]. On the platform of linear optics, the problem of quantum state amplification has been addressed both theoretically and experimentally. In 008, Ralph and Lund introduced the concept of probabilistic noiseless linear amplification of quantum states [104]. Two years later, several research groups have independently reported on an experimental heralded amplification of an attenuated coherent state of light [ ]. Subsequently, N. Gisin et al. proposed a scheme for amplification of photon polarisation qubits [105] which was experimentally tested in 01 by S. Koscis et al. [111]. This scheme however suffers from asymptotically vanishing success probability for infinite values of gain (corresponding to complete vacuum removal). To address this issue, D. Pitkanen et al. [11] and M. Curty with T. Moroder devised improved versions of this scheme [113]. 6.1 Entanglement-based linear-optical qubit amplifier Commenting on E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein and K. Lemr, Entanglement-based linear-optical qubit amplifier, Phys. Rev. A 88, 0137 (013) [A9]. In 013, we proposed yet another amplification scheme for photon polarisation qubits [A9]. Our goal was to overcome various shortcomings of the previous proposals. Firstly, success probability of our scheme does not vanish for infinite gain like in the N. Gisin et al. scheme [105]. Further to that, in the infinite gain regime, our scheme does not require photon-number resolving detectors which is the case of the D. Pitkanen et al. scheme [11]. And in contrast to the Curty- Moroder scheme [113], we can tune the amplification gain by setting parameters of individual components.

57 6.1. ENTANGLEMENT-BASED LINEAR-OPTICAL QUBIT AMPLIFIER 45 Figure 6.1: Scheme for a linear-optical qubit amplifier as described in the text. EPR source of entangled ancillary photon pairs, PBS polarising beam splitter, PPBS partially polarising beam splitter (defined in the text), WP wave plate, PDF polarisation dependent filter, D polarisation resolving detection. Linear-optical setup of our amplifier is depicted in Fig We assume the input signal state to be a superposition of vacuum and single photon polarisation state ψ in = α 0 + β H H + β V V, (6.3) where 0 denotes the vacuum component while H and V stand for horizontally and vertically polarised single photon states respectively. The coefficients in Eq. (6.3) follow a normalisation condition α + β H + β V = 1. In the first step, the input signal polarisation modes are spatially split on a polarising beam splitter PBS in. Subsequently, each of these modes impinge on two partially polarising beam splitters PPBS1 and PPBS, where they interact with one photon of an ancillary Bell state. One of these partially polarising beam splitters completely transmits horizontally polarised light and reflects vertically polarised light with amplitude reflectivity r. The second partially polarising beam splitter completely transmits vertically polarised light and reflects horizontally polarised light with an identical reflectivity factor r. Finally, the signal state polarisation modes are recombined on the output polarising beam splitter PBS out. The amplifier succeeds if one photon is detected in ancillary output mode of each partially polarising beam splitter (detectors D1 and D). Depending on the polarisation measurement on these ancillary modes, a feed-forward operation is imposed to the output signal state. Upon a successful operation, the signal state

58 46 CHAPTER 6. LINEAR-OPTICAL QUBIT AMPLIFIERS Figure 6.: Amplification success rate as a function of the gain G depicted for three different input states characterised by the vacuum probability α. is transformed into ψ out αr 0 + 3r 1 (β H H + β V V ). (6.4) The polarisation photon state is thus amplified with a gain factor G depending on the reflectivity coefficient r G = 3r 1 4r. (6.5) Success probability of the entire amplification procedure depends on the parameters of the input state as well as on the imposed gain P succ = r α + G β H + β V. (6.6) We have calculated the success probability P as a function of the gain G for various input states parametrised by the vacuum amplitude α. Results of these calculations are visualised in Fig. 6.. Couple months later, A. Máttar et al. proposed a similar scheme for spincoupled cavities [114]. In 015, our scheme has been generalised by Tie-Jung Wang and Chuan Wang to operate on entangled photon pairs instead of single photons [115]. 6. State-dependent linear-optical qubit amplifier Commenting on K. Bartkiewicz, A. Černoch, and K. Lemr, State-dependent linearoptical qubit amplifier, Phys. Rev. A 88, (013) [A10]. Heralded amplifiers for photon polarisation qubits discussed in the preceding section operate noiselessly, thus following the condition ˆρ = ˆρ (see Eq. 6.).

59 6.. STATE-DEPENDENT LINEAR-OPTICAL QUBIT AMPLIFIER 47 (a) (b) (c) (d) Figure 6.3: Success probability P succ as a function of both output state fidelity F and amplification gain G is depicted for four different input states: (a) β H / β V = 1, (b) β H / β V = 0.73, (c) β H / β V = 0.58, (d) β H / β V = THR denotes the threshold delimiting the region of inaccessible combinations of fidelity and gain. In our second paper on qubit amplification [A10], we have considered relaxing this requirement in order to improve the success probability of the amplification procedure. Since the output qubit state may no longer be identical to the input qubit state, we have to introduce fidelity of the amplification procedure F ˆρ, ˆρ = Tr ˆρ ˆρ ˆρ, (6.7) which reaches a value of 1 for perfect noiseless amplification. For the purposes of this research, we have modified our scheme for noiseless amplifier discussed in the preceding section [A9], but generalised the ancillary state to ψ a = cos χ HH + sin χ V V. (6.8) We have started by calculating the maximum success probability as a function of both the output state fidelity and amplification gain. Fig. 6.3 shows the obtained results for four different input state parametrised by the ratio β H /β V. It follows from our calculations, that in the infinite gain regime one can improve the success rate at the expense of lower fidelity. This is however not generally true for finite values of gain.

60 48 CHAPTER 6. LINEAR-OPTICAL QUBIT AMPLIFIERS In the next step, we have considered the amplification procedure operating with some a priori information about the input qubit state. The exactness of the a priori information was characterised by the concentration parameter κ of the Mises-Fisher distribution on the Bloch sphere g(θ, κ) = κ exp (κθ), (6.9) 4π sinh(κ) where tan θ = β V /β H. We have then calculated the relation between success probability and average fidelity for various values of concentration parameter κ and average gain G. Interpretation of our results is most straightforward for infinite gain. With increasing exactness of the a priory information about the input qubit, the amplifier shifts from the qubit amplification regime to simpler photon amplification regime. This is accompanied by increasing success probability for all values of average fidelity smaller then 1. In an extreme case of exact knowledge of the input qubit state, the infinite gain success probability reaches its maximum value of (1 α ).

61 Chapter 7 Conclusions In this thesis, I have summarised my research on the tunable c-phase gate [A11] and related topics. The c-phase gate represents a key ingredient for quantum information processing and its investigation thus contributes to further development of this scientific field. The tunability of the gate s phase shift increases even more its capabilities and versatility. Our team has investigated both the fundamental properties of the tunable c-phase gate (Chapter 3) as well as its applications (Chapter 4). In 01, we identified causes for the non-monotonic dependence of the gate s success probability on its phase shift [A1]. This work sheds light on the physical processes taking place inside the gate and explains its success probability in terms of conditions imposed on the gate s operation (see Sec. 3.1). Also in 01, we analysed the entangling properties of the tunable c-phase gate [A]. We have put forward a new measure of entangling capabilities of quantum gates the entangling efficiency. This new measure establishes a more exact way to compare probabilistic quantum gates (see Sec. 3.). The research deepens our understanding of the tunable c-phase gate. It allows us to fully appreciate the benefits of the tunable phase shift, a necessary prerequisite to make full use of this gate in quantum information processing. In 011, first application of the optimal tunable c-phase gate was proposed. The scheme for preparation of the so-called KLM states makes use of the specific form of its success probability function to maximise the overall success rate [A13] (see Sec. 4.1). KLM states can serve as ancillae for more efficient quantum information processing on the platform of linear optics. Preparing them is the first step towards their application. Subsequently, we have focused on the topic of quantum routing (see Sec. 4.). We have devised two schemes for linear-optical quantum routers [A4, A5]. Our second scheme explicitly involves the tunable c-phase gate. Quantum routers steer signal qubits in a quantum network depending on the state of control qubits. They allow to develop more sophisticated quantum communications networks connecting multiple users. Establishing how to implement quantum routers on the platform of linear optics increases the potential of using 49

62 50 CHAPTER 7. CONCLUSIONS discrete photons in future complex quantum communications networks. Our research further promoted quantum routers by presenting two additional tasks they can undertake: quantum message authentication and Bell state manipulation [A6] (see Sec. 4.3). Perhaps the most important result presented in this thesis is the construction of controlled-unitary gates using the tunable c-phase gate [A7]. We have demonstrated a considerable improvement when standard CNOT gates are replaced by tunable c-phase gates (see Sec. 4.4). Moreover, we have constructed one such controlled-unitary gate and experimentally verified our idea. The capability to efficiently implement various controlled-unitary gates is crucial for quantum circuit design especially on the platform of linear optics. We have explicitly shown how to achieve such task with unprecedented efficiency. In order to make the c-phase gate even more versatile, we have proposed a scheme for programmable version of this gate [A8]. Instead of tuning the phase shift by settings of various linear-optical components, a program qubit is applied instead (see Chapter 5). The phase shift that the gate imposes is encoded directly in this program qubit state. Programmable quantum gates allow to transmit entire quantum information protocols in the form of a string of program qubits. Since single-qubit programmable gates had already been analysed, we have filled the missing piece in the universal toolbox by proposing a scheme for programmable c-phase gate. While the c-phase gate allows to implement various logical operations on qubits, their transmission across large distances requires efficient amplification schemes. In Chapter 6, our scheme for linear-optical qubit amplifier [A9] is discussed. This scheme is compared to previously known linear-optical qubit amplifiers and its benefits pointed out. Subsequently, our amplification procedure is generalised allowing some level of noise. Trade-off between output state fidelity and amplification success rate is then investigated [A10]. Our amplifier outperforms previously known schemes achieving higher efficiency of this important quantum communications protocol. The general goal of the presented research is to contribute to further development of quantum information science. The platform of discrete photons and linear optics seams particularly suitable for construction of quantum communications networks. Devices like quantum routers and qubit amplifiers are indispensable for practical realisation of these networks. As it has been demonstrated, the tunable c-phase gate can be an invaluable resource for implementation of such devices or other related quantum information protocols. 7.1 Future plans It is imperative to subject theoretical proposals to experimental implementation. This way functionality of these proposals is verified and their feasibility tested with real-life components and tools. Some of the ideas presented in this thesis

63 7.1. FUTURE PLANS 51 have already been experimentally realised. My future plans include experimental realisation of the remaining ones. Especially, I plan to construct the programmable c-phase gate and our quantum amplification scheme. Besides the research related to the topic this thesis, we plan on investigating some fundamental aspects of quantum optics. We are about to perform a series of four-photon experiments testing various entanglement witnesses and entanglement quantification schemes. Preliminary results of this research activity have already been sent to ArXiv [A16]. The planned research should provide new insights into fundamental quantum phenomena which in turn can be used for practical quantum information processing. Such investigation is important for two reasons: firstly it expands our understanding of nature and secondly it gives us means of how to directly benefit from this knowledge. I believe that even at the bachelor s or master s level, students can contribute to serious research activities. I try, as much as possible, to offer my students topics for their theses that can be subsequently published in scientific journals. Some of my students have thus already published their first papers [A9, A14, A17]. I plan on continuing in this practice and offer my students to participate on the above envisioned topics.

64 5 CHAPTER 7. CONCLUSIONS

65 Author s publications Commented on in this thesis (see Supplementary Material) [A1] [A] [A3] [A4] [A5] [A6] [A7] [A8] [A9] [A10] K. Lemr and A. Černoch, Optimal success probability of a tunable linearoptical controlled-phase gate, Phys. Rev. A 86, (01). K. Lemr, A. Černoch, J. Soubusta, and M. Dušek, Entangling efficiency of linear-optical quantum gates, Phys. Rev. A 86, 0331 (01). K. Lemr, Preparation of Knill Laflamme Milburn states using a tunable controlled phase gate, J. Phys. B: At. Mol. Opt. Phys. 44, (011). K. Lemr and A. Černoch, Linear-optical programmable quantum router, Opt. Comm. 300, 8 85 (013). K. Lemr, K. Bartkiewicz, A. Černoch, and J. Soubusta, Resource-efficient linear-optical quantum router, Phys. Rev. A 87, (013). K. Bartkiewicz, A. Černoch, and K. Lemr, Using quantum routers to implement quantum message authentication and Bell-state manipulation, Phys. Rev. A 90, 0335 (014). K. Lemr, K. Bartkiewicz, A. Černoch, M. Dušek, and J. Soubusta, Experimental Implementation of Optimal Linear-Optical Controlled-Unitary Gates, Phys. Rev. Lett. 114, (015). K. Lemr, K. Bartkiewicz, and A. Černoch, Scheme for a linear-optical controlled-phase gate with programmable phase shift, J. Opt. 17, 150 (015). E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein, and K. Lemr, Entanglement-based linear-optical qubit amplifier, Phys. Rev. A 88, 0137 (013). K. Bartkiewicz, A. Černoch, and K. Lemr, State-dependent linear-optical qubit amplifier, Phys. Rev. A 88, (013). 53

66 54 AUTHOR S PUBLICATIONS Other, related to this thesis [A11] [A1] [A13] [A14] [A15] [A16] [A17] K. Lemr, A. Černoch, J. Soubusta, K. Kieling, J. Eisert, and M. Dušek, Experimental implementation of the optimal linear-optical controlled phase gate, Phys. Rev. Lett. 106, 1360 (011). K. Lemr and J. Fiurášek, Preparation of entangled states of two photons in several spatial modes, Phys. Rev. A 77, 0380 (008). K. Lemr, A. Černoch, J. Soubusta, and J. Fiurášek, Experimental preparation of two-photon Knill-Laflamme-Milburn states, Phys. Rev. A 81, 0131 (010). M. Bula, K. Bartkiewicz, A. Černoch, and K. Lemr, Entanglement-assisted scheme for nondemolition detection of the presence of a single photon, Phys. Rev. A 87, (013). K. Bartkiewicz, A. Černoch, K. Lemr, J. Soubusta, and M. Stobińska, Efficient amplification of photonic qubits by optimal quantum cloning, Phys. Rev. A 89, 063 (014). K. Lemr, K. Bartkiewicz, and A. Černoch, Experimental measurement of the collectibility of two-qubit states, arxiv: (016). K. Bartkiewicz, J. Beran, K. Lemr, M. Norek, and A. Miranowicz, Quantifying entanglement of a two-qubit system via measurable and invariant moments of its partially transposed density matrix, Phys. Rev. A 91, 033 (015).

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75 Supplementary Material S-1

76 S-

77 PHYSICAL REVIEW A 86, (01) Optimal success probability of a tunable linear-optical controlled-phase gate Karel Lemr * and Antonín Černoch RCPTM, Joint Laboratory of Optics of Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic, 17. listopadu 1, Olomouc, Czech Republic (Received 16 July 01; published 5 September 01) In this Brief Report we provide the physical interpretation of the success probability of an optimal tunable linear-optical controlled-phase gate as a function of the phase shift imposed by the gate. We analyze in detail the theoretical prediction of this success probability as identified by Kieling et al. [New J. Phys. 1, (010)] and provide discussion revealing the reasons for its nonmonotonous behavior. Moreover, we show that the success probability can be increased if several requirements imposed on the gate can be lifted. DOI: /PhysRevA PACS number(s): Lx, 4.50.Dv Quantum information processing with linear optics is a dynamically developing field of research. Over the past decades, theoretical and experimental investigation have delivered a number of important protocols implementing key features of quantum computing and communication [1 3]. An important building block for these protocols is the controlled-phase gate (or controlled-not gate) [4 9]. This two-qubit gate makes part of the universal set of gates that can be used to implement any QIP device [1,9]. Moreover, this gate also finds its application in a number of prominent quantum information protocols such as quantum Fourier transform [10,11], quantum teleportation [1], entanglement distillation [13,14], or quantum nondemolition measurement [15]. The most important parameter of a controlled-phase gate (CPHASE gate) is the phase shift ϕ that the gate imposes between logical states 0 and 1 of the signal qubit if the control qubit issetto 1. Previously implemented gates were mainly limited to fixed value of phase shift ϕ = π [6,7]. In contrast to that a CPHASE gate is said to be tunable if the phase shift can be varied in the interval ϕ [0,π] [16]. Similar to many other linear-optical gates, the tunable CPHASE gate is inherently probabilistic. With a given success probability, the gate succeeds, which can be witnessed by a presence of signal and control photons in their respective output ports (dual-rail encoding). On the other hand, the gate fails if either one or both of the photons are lost due to the filtration in the setup or if the photons bunch, leaving the gate by the same output port. This behavior of the gate prevents its direct scalability. If the gate is used only once, a postselection on simultaneous presence of photons in control and signal output ports has to be used to select only successful cases. In order to use the gate repeatedly, one has to implement a quantum nondemolition measurement inserted to the output of each gate. Note that all-linear-optical implementation of such quantum nondemolition measurement has already been presented [15]. In a recent paper, Kieling et al. have identified the success probability of an optimal tunable linear-optical CPHASE gate as a function of imposed phase shift [17]. Although they have established the theoretical formula for its success probability * lemr@jointlab.upol.cz in the form of [ P (ϕ) = 1 + sin ϕ ( ) π ϕ sin + 3/ ϕ sin 1/] 4 (1) (where ϕ denotes the phase imposed by the gate), the physical interpretation for this relation has not yet been given. Kieling et al. have only determined that the implementation requires additional ancillary mode to impose phases other then 0 or π. Besides this observation, the reason for the success probability to be quite surprisingly nonmonotonous function of the phase shift was never explained, even though it has been experimentally verified [16]. Nor have the properties of the success probability been analyzed based on the optical process of the two-photon interference. It is interesting from both the conceptual and technical point of view to address this problem more closely by providing the physical interpretation of the gate functioning and especially the reasons for the specific form of the success probability function. In this paper, we provide detailed analysis of the success probability of optimal CPHASE gate with particular emphasis on the reasons for its nonmonotonous behavior. The gate s operation can be expressed in terms of its computational basis, e iϕ 11, () where numbers 0 and 1 denote logical states of signal and control qubit. Schematic depiction of the optimal linear-optical implementation of the CPHASE gate is provided in Fig. 1. One can observe that both the signal and control modes corresponding to logical states 0 are just subjected to local phase shifts α s and α c for signal and control qubit, respectively, and, if necessary, to filtering with amplitude transmissivities τ S and τ C. These modes do not undergo any two-mode interaction. The 1 signal and control modes, however, interact via a Mach-Zehnder interferometer with one arm subjected to tunable amplitude losses τ and the other arm to phase shift α I. Because of the symmetry of the gate, there are three different paths the qubits can take. These are: (I) the input state 00, where no qubit goes through the interferometer; /01/86(3)/034304(4) American Physical Society S-3

78 BRIEF REPORTS PHYSICAL REVIEW A 86, (01) control signal { { BS FIG. 1. Schematic depiction of the linear-optical implementation of a tunable CPHASE gate. The input states (on the left-hand side) are transformed by the optical components of the gate to the output state (on the right-hand side). Only those cases, when there is exactly one signal and one control qubit at the output are considered as successful. Note that this is achieved by postselection on coincidences in the real experimental setup. The labels α S, α C,andα I denote phase shifts; τ, τ S,andτ C denote tunable neutral density filters; BS denotes balanced beam splitters. (II) states 01 and 10, where exactly one qubit enters the interferometer; and (III) state 11, where both qubits interact in the interferometer. Case (I) is trivial, its transformation reads S I C BS 00 τ S τ C e i(αs+αc) 00. (3) Let us now derive the transformation equation for case (II) exactly one qubit, signal, or idler, entering the interferometer. The prescription Eq. () requires that for successful operation, the qubit has to leave the interferometer by the same output port. Signal input qubit has to leave the setup by signal output port and control input qubit by control output port. The opposite case shall be disregarded and, hence, lowers the success probability. The above-mentioned notes lead to formulating the transformation equation in the form of 01 A 1 τ S e iαs A 1 τ C e iαc 10, A 1 = 1 (τ + ), (4) eiαi where only the successful cases (exactly one photon in each output port) have be taken into account. 1 The overall phase shift imposed on states 01 and 10, reads (α s + α 1 ) and (α c + α 1 ), respectively, where α 1 stands for the phase shift that arises from the amplitude A 1 α 1 = (A 1 ) arctan Im(A 1) Re(A 1 ). (5) In the case (III) ( 11 input state), there are two-qubits entering the interferometer at once. In this case, the successful gate operation is observed, when both qubits leave the interferometer by separate outputs. Simple calculation reveals the transformation 11 A 11, A = 1 (τ + e iαi ) (6) and similarly to the previous situation, one can calculate the phase shift imposed in this case α = (A ). 1 For a more convenient calculation, in our notation the beam splitter implements a two-mode transformation described by the matrix 1 (1 1; 1 1), where the top mode from scheme in Fig. 1 is considered always as the first mode. Note that the final results are not affected by this choice of beam splitter formalism. S C A 1, A A 1, A (a) A 1 A /4 / /4 I [rad] (b) /4 / /4 I [rad] A 1 A FIG.. Determining the working points for (a) τ = 1 (corresponding gate phase shifts ϕ ={0,π}) and (b) τ = 0.6 (corresponding gate phase shifts ϕ {0.38π,0.05π}). The phase in the interferometer α I has to be set so that the condition A 1 4 = A is fulfilled. The phases satisfying this condition are visualized by vertical dashed lines. The gate operation prescription Eq. () imposes several conditions on the phase shifts α S, α C, α 1, and α : α S + α 1 = α C + α 1 α C = α S α S + α C = α C + α 1 α S = α 1 (7) α (α 1 + α S ) = ϕ ϕ = α α 1. Besides the phase shifts condition, there is yet another requirement concerning the success probability of the gate. In order to operate correctly, the gate needs to balance success probabilities related to all four computational basis states. If that would not be so, the gate would transform incorrectly general superpositions of basis states. To illustrate this issue, suppose the transformation of the following state: 1 ( ) 1 ( P P ). (8) If probabilities P 00 and P 10 differ, the output state would be deformed and would no longer match the desired output state of the form of P ( ). (9) This analysis reveals the following requirement imposed on the success probabilities of transforming the four basis states: P 00 = P 01 = P 10 = P 11, (10) S-4

79 BRIEF REPORTS PHYSICAL REVIEW A 86, (01) 7 /8 3 /4 5 /8 / 3 /8 /4 /8 0 I FIG. 3. Values of working points α I (full line) and gate phase shifts ϕ (dotted line), both as a functions of filtering τ. where indexes denote the corresponding states. Expressing these success probabilities using Eq. (4), Eq.(6), and filter transmissivities τ S and τ C, one obtains the resulting equations that the gate parameters have to observe: τc τ S = τ C A 1 τs = A 1 τc τ S = τ S A 1 τc = A 1 (11) τc τ S = A A 1 4 = A. To summarize the performed calculations, there are two conditions to be fulfilled: the phase shift condition Eq. (7) and the success probability condition Eq. (11). Both the conditions have to be fulfilled simultaneously. Let us now focus on the second condition. Since A 1 and A are sinus functions of the interferometer phase shift α I, it is easy to verify that for any amount of filtering τ there exist at maximum two solutions (to be referred as working points) satisfying the third line in Eq. (11). For any fixed value of τ, the phase in the interferometer has to be set according to the quadratic equation 1τ C 4(1 + τ )τc + 3(1 + τ 4 ) 10τ = 0, (1) where C = cos α I. The specific working points α I fulfilling the success probability condition Eq. (11) are found by solving this equation yielding the solution cos α I = 1 + τ ± τ (τ 1). (13) 6τ Figure illustrates the position of these working points for two amounts of filtering τ = 1 and τ = 0.6. Generally, the two working points get closer to each other when increasing the level of filtration (decreasing transmissivity). Figure 3 witnesses this trend. Also note that there exist two solutions to the quadratic Eq. (1) for 1 τ> 1 ( 3 1) For τ = 1 ( 3 1), there exists only one solution, and for values of τ< 1 ( 3 1), Eq. (1) has no physical solution. In order to determine the phase shift imposed by the gate in any of the valid working points, one uses the Eqs. (7). Examining the phase shift imposed by the gate at every working point confirms the previously known fact, that phase shifts 0 and π are obtained without filtering in the interferometer (τ = 1) and that further tuning of the filter τ offers the possibility of reaching any other gate phase shift ϕ (see Fig. 3). FIG. 4. Graphical visualisation of the trade-off as described in the text. Note that there are two competing effects: distance between the working point α I and the minimum of interference fringe at π/ (dotted line) and the actual value at that minimum (dashed line). This trade-off leads to global minimum of the success probability (full line) of the gate for phase shift ϕ 0.58π. The analysis carried out in the preceding paragraphs also reveals the reason for the nonmonotonous behavior of the success probability. There are two combined effects that form the trade-off responsible for that behavior. First, one can observe from Fig. that in order to decrease the gate phase shift from π to lower values, (a) the corresponding amount of filtering has to be increased, and, additionally, (b) the working point α I approaches π/ where A realizes its minimal value [compare Figs. (a) and (b)]. The effects (a) and (b) form the trade-off. Because of the effect (a) (increased filtering), the visibility of interferometric fringes decreases and the minimum value of A increases. However, the effect (b) (working point gets closer to the minimum at π/) leadsto a drop in the success probability. Figure 4 depicts this trade-off between the effects (a) and (b) by presenting the value A at its minimum and the distance of the working point to that minimum, both as functions of the gate phase shift ϕ. Corresponding success probability is also depicted. Note that the seemingly intuitive fact that the minimum success probability would be at the working point α I = π/ isfalse. The trade-off between the mentioned effects leads to minimize the overall success probability at α I 0.58π, where the gate implements phase ϕ 0.61π. In this Brief Report, we have provided detailed insight into the success probability of an optimal tunable linear-optical CPHASE gate. We have presented arguments explaining the nonmonotonous behavior of the success probability as a function of the imposed phase shift ϕ. This analysis revealed that the main limiting factor for the success probability is the requirement imposed by the condition of Eq. (11). There are quantum information schemes, such as the one for quantum nondemolition measurement [15], that do not require a full featured CPHASE gate. In these cases, the condition of Eq. (11) can be dropped, opening space for increased success probability of the scheme in question S-5

80 BRIEF REPORTS PHYSICAL REVIEW A 86, (01) The authors thank Jan Soubusta for fruitful discussion on the subject of this paper. The authors also acknowledge the support by the Operational Program Research and Development for Innovations European Regional Development Fund (Project No. CZ.1.05/.1.00/ ) and the Operational Program Education for Competitiveness European Social Fund (Projects No. CZ.1.07/.3.00/ and No. CZ.1.07/.3.00/0.0058) of the Ministry of Education, Youth and Sports of the Czech Republic. [1] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 00). [] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, P. Jonathan, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (007). [3] I. A. Walmsley, Science 319, 111 (008). [4] A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, Phys.Rev.Lett. 74, 4083 (1995). [5] H. F. Hofmann and S. Takeuchi, Phys. Rev. A 66, (00). [6] T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D. Franson, Phys. Rev. A 68, (003). [7] N. Kiesel, C. Schmid, U. Weber, R. Ursin, and H. Weinfurter, Phys. Rev. Lett. 95, (005). [8] M. Bartkowiak and A. Miranowicz, J. Opt. Soc. Am. B 7, 369 (010). [9] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Phys. Rev. A 5, 3457 (1995). [10] Michael Clausen, Theoret. Comput. Sci. 67, 55 (1989). [11] L. Hales and S. Hallgren, FOCS 00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Washington, DC, USA, 000), p [1] L. Roa, A. Delgado, and I. Fuentes-Guridi, Phys. Rev. A 68, 0310 (003). [13] J.-W. Pan, C. Simon, C. Brukner, and A. Zeilinger, Nature (London) 410, 1067 (001). [14] G. Alber, A. Delgado, N. Gisin, and I. Jex, J. Phys. A 34, 881 (001). [15]G.J.Pryde,J.L.O Brien,A.G.White,S.D.Bartlett,andT.C. Ralph, Phys. Rev. Lett. 9, (004). [16] K. Lemr, A. Černoch, J. Soubusta, K. Kieling, J. Eisert, and M. Dušek, Phys.Rev.Lett.106, (011). [17] K. Kieling, J. L. O Brien, and J. Eisert, New J. Phys. 1, (010) S-6

81 PHYSICAL REVIEW A 86, 0331 (01) Entangling efficiency of linear-optical quantum gates Karel Lemr, 1,* Antonín Černoch, 1 Jan Soubusta, 1 and Miloslav Dušek 1 Institute of Physics of Academy of Sciences of the Czech Republic, Joint Laboratory of Optics of PU and IP AS CR, 17. listopadu 50A, Olomouc, Czech Republic Department of Optics, Faculty of Science, Palacký University, 17. listopadu 1, Olomouc, Czech Republic (Received 4 July 01; published 17 September 01) We propose a measure of the nonclassicality of quantum gates which is particularly suitable for probabilistic devices. This measure enables to compare deterministic devices which prepare entangled states with a low amount of entanglement with probabilistic devices which generate highly entangled states but which fail sometimes. We provide examples demonstrating the advantages of this measure over the so-far employed entangling power. The experimentally determined entangling efficiency of a tunable linear-optical controlled-phase gate is presented to highlight the practical aspects of this measure. DOI: /PhysRevA PACS number(s): Bg, 4.50.Ex, Lx I. INTRODUCTION Quantum physics has opened new ways in information theory. Quantum computing and quantum information processing have attracted great attention in the last few decades both in the theoretical and experimental domains [1]. Quantum gates, the devices analogous to classical logical gates, are necessary ingredients for building quantum circuits. A considerable effort was devoted to implement them experimentally on several physical platforms that represent good candidates for these tasks [ 5]. One of them is linear optics. The advantages of linear-optical quantum gates are that they are accessible by a present-day technology, their realization is relatively simple [6], there is a chance of their integration [7], and they work directly with light so they are convenient for informationprocessing tasks connected with quantum communication [8]. The disadvantage is that they are mostly probabilistic (i.e., they operate with the success probability lower than 1). This problem can be overcome by more complex setups based on prearranged entangled states [9,10]. However, even the basic linear-optical quantum gates with success probability lower than 1 can be quite useful for small-scale applications of quantum information processing. This is especially true for the circuits behind or between quantum links because in real-life quantum channels huge losses must be tolerated anyway. It is an interesting question how to quantify the performance of quantum gates. It was proposed to use the so-called entangling power [11 13]. This measure is defined either as an average or maximal value of the measures of entanglement of the output states over all separable input states. Of course, quantum gates are not primarily intended as entanglement sources (the sources of entangled states can be implemented more easily [14]), but their capability to create entanglement from separable input states is crucial for quantum information processing. Thus entangling power is a good measure of the nonclassicality of quantum gates. However, this measure is disputable in the case of probabilistic quantum gates. One can consider two distinct cases: (i) a deterministic device which prepares entangled states with a low amount of entanglement and (ii) a probabilistic * lemr@jointlab.upol.cz device which generates highly entangled states but which fails sometimes. What is better? We can imagine we have a perfect entanglement-distillation apparatus which we apply to the output of the first device (i). We use it to obtain (asymptotically) such a number of distilled states which equals to the number of states generated by the second device (ii). The number of input states is assumed to be the same for both devices. Now, having the same fraction of states per a gate operation, we can find which of the two cases (i or ii) leads to a higher amount of entanglement. Thus a good measure of the performance of probabilistic quantum gates could be the product of distillable entanglement [15,16] and success probability maximized over all separable input states. But there are two problems with this definition. The first one is practical: It is difficult to calculate distillable entanglement for a general state. The second one is conceptual: This definition fails to quantify the ability to generate bound entangled states. Therefore it is convenient to generalize the definition of this quantity as the maximum (over all separable input states) of the product of success probability and any well-behaved entanglement measure chosen according to one s particular needs. We will call this function entangling efficiency. It is reasonable to choose an entanglement measure which is convex because then the maximum can be taken only over all pure product input states. In the following text we will define the particular form of entangling efficiency using negativity as the measure of entanglement and we will use it to characterize the optimal linear-optical controlled phase gate recently implemented in our laboratory. II. ENTANGLING EFFICIENCY In this paper we will measure the amount of entanglement in a quantum state by its negativity. Negativity of state ρ is defined as [17] N(ρ) = ρta 1 1, (1) where 1 denotes the trace norm and T A means the partial transpose. This measure can be easily calculated. It is convex, N( i p iρ i ) i p in(ρ i ), and it is an entanglement monotone (does not change under local operation and classical /01/86(3)/0331(5) American Physical Society S-7

82 LEMR, ČERNOCH, SOUBUSTA, AND DUŠEK PHYSICAL REVIEW A 86, 0331 (01) communication). However, it is zero even if the state is entangled under positive partial transpose. Entangling power is defined as the supremum of the negativity of output states over all separable input states E p = sup{n(e[ρ])}, () ρ S where E[ρ] denotes the output state of the device corresponding to input state ρ (E is a completely positive map) and S is the set of all separable input states. The convexity of negativity is important because for a nonconvex entanglement measure there might be a mixed state whose measure of entanglement is greater than the negativity of any of the pure states of which it is composed. This fact would complicate the search for the input state corresponding to the maximum because mixed states have significantly more degrees of freedom than pure states. We define entangling efficiency in the following way: E eff = sup{p s (ρ)n(e[ρ])}, (3) ρ S where p s (ρ) is the success probability of the gate for a given input state ρ. The probabilistic operation of linear optical quantum devices means that only some of their output states correspond to the correct results. However, we know which ones and we can select them either by an auxiliary measurement or by postselection. In all the examples considered below, the successful operation of a gate corresponds to the cases where both photons leave the gate separately by their respective output ports. In the experiment, these cases are selected by coincidence detection. All the other cases (e.g., two photons appearing in the same output port) are considered as gate failures. III. EXAMPLE 1: ENTANGLING EFFICIENCY OF A BEAM SPLITTER We illustrate the concept of entangling efficiency on an intuitive case of a beam splitter followed by a postselection. Let us begin with a balanced beam splitter with transmittance T and reflectance R both equal to 1. Two photonic qubits are initially in a separable state and each of them enters one input port of the beam splitter. Subsequently we perform a postselection taking into account only the cases where there is exactly one photon in each output mode. Let us express a separable input state ψ 1 ψ using the following parametrization (without the loss of generality one part of the input state can be fixed): ψ 1 = 0, ψ =cos θ 0 +e iϑ sin θ 1, (4) where { 0, 1 } represents an arbitrary orthogonal basis. Simple algebra reveals that the success probability of the above-described beam-splitter transformation reads p s (θ) = sin θ. (5) Using definition (1) one can calculate the negativity of output states N = 1, (6) max {p s } max {Np s } 0 /4 / 3 /4 Angle [rad] p s N Np s FIG. 1. Balanced beam splitter. Success probability p s (full line), negativity N (dotted line) and their product (dashed line) as functions of angle θ parametrizing the input separable state. which is independent of the input state. Therefore the entangling power of the beam splitter reads E p = 1. (7) As for the entangling efficiency, one has to find the maximum of the product of success probability and negativity Np s = sin θ. (8) 4 This product is maximized for θ = π/ so the entangling efficiency is E eff = 1 4. (9) Figure 1 shows the dependence of success probability p s, negativity N, and their mutual product as functions of angle θ. Finding the maximum of negativity N reveals the entangling power E p of the beam splitter, whereas finding the maximum of the product Np s reveals the entangling efficiency E eff.note that in this case both of them are maximized for the same input state parameter θ. One can extend the model of the balanced beam splitter also to the case of a general lossless beam splitter. We have calculated the success probability p s, entangling power E p, and entangling efficiency E eff also for this case. Figure p s E p E eff Transmitance T FIG.. Unbalanced beam splitter. Success probability p s (full line), entangling power E p (dotted line), and entangling efficiency E eff (dashed line) of a beam splitter as functions of its transmittance T S-8

83 ENTANGLING EFFICIENCY OF LINEAR-OPTICAL... PHYSICAL REVIEW A 86, 0331 (01) presents these three properties of the beam splitter as functions of its transmittance. IV. EXAMPLE : OPTIMAL LINEAR-OPTICAL C-PHASE GATE We further demonstrate our concept of entangling power on the second example: The optimal linear-optical controlledphase (c-phase) gate. A controlled-phase gate implements the following operation on two qubits: 0,0 0,0, 0,1 0,1, (10) 1,0 1,0, 1,1 e iϕ 1,1. In general, it is an entangling quantum gate. Together with single-qubit operations it forms a universal set for quantum computing. For example, the controlled-not gate can be obtained by applying a Hadamard transform to the target qubit before and after the controlled-phase gate with phase shift π. Recently, we built the optimal linear-optical controlled phase gate in our laboratory [18]. Its conceptual scheme is depicted in Fig. 3. Phase shift ϕ applied by this gate on the controlled qubit can be set to any given value just by tuning the parameters of the setup. The gate is optimal in the sense that for any phase shift it operates at the maximum possible success probability that is achievable within the framework of any postselected linear-optical implementation without auxiliary photons. The optimal success probability of the gate takes the following form [19] ( p s (ϕ) = 1 + sin ϕ + 3/ sin π ϕ 4 sin ϕ 1/). (11) The dependence of the success probability on the phase shift is shownonfig.4. Surprisingly it is not monotone in the phase. To evaluate the entangling power and efficiency of this gate, let us express separable input state ψ 1 ψ using the following parametrization ψ 1, =cos θ 1, 0 +e iϑ1, sin θ 1, 1, (1) FIG. 3. Scheme of the gate [18]. Vertically (V ) and horizontally (H ) polarized components of the same beam are drawn separately for clarity. In polarization beam splitters PBS1 and PBS the vertical components are reflected. Half-wave plates HWPb and HWPc act as beam splitters for V and H polarization modes. F1 and F are filters (attenuators), F1 acts on both polarization modes, F on the H component only. Phase shifts φ + and φ are introduced by proper path differences in the respective modes. HWPa and HWPd just swap vertical and horizontal polarizations. In the final setup they are omitted for simplicity and the second qubit is encoded inversely with respect to the first qubit p s (left axis) E p (left axis) E eff (right axis) 0 /4 / 3 /4 Gate phase shift FIG. 4. C-phase gate. Success probability p s (full line), entangling power E p (dotted line), and entangling efficiency E eff (dashed line) as functions of phase shift ϕ applied by the gate. Theoretical prediction is depicted by lines whereas the experimentally obtained data are depicted using markers. where { 0, 1 } represents a fixed computational basis and the indices denote the first and second qubits. Further, let us assume this state is successfully transformed by the gate according to Eq. (10). Then we can calculate the negativity of the output two-qubit state using Eq. (1) N(ϕ,θ 1,θ ) = sin θ 1 sin θ (1 cos ϕ), (13) where ϕ denotes the phase applied by the gate. One can notice that this function does not depend on phases ϑ 1, but only on θ 1, [0, π ]. For any given value of gate phase shift ϕ, it is maximized for θ 1, = π (equal superposition 4 of computational basis states). The maximum negativity and therefore the entangling power for a given phase ϕ thus reads E p (ϕ) = 1 cos ϕ. (14) 4 Similarly, the entangling efficiency can be obtained by maximizing the product N(ϕ,θ 1,θ ) p s (ϕ) over the input-state parameters θ 1,. Because the success probability does not depend on the input state, we obtain the entangling efficiency as a function of the gate phase shift ϕ in the form ps E eff = 1 cos ϕ. (15) 4 Clearly, the success probability of our c-phase gate is state independent [see Eq. (11)], so the entangling efficiency of the gate is just a product of the success probability and entangling power. In Fig. 5 there are plots of success probability p s, negativity N, and their product in dependence on parameter θ 1 for ϕ = π and θ = π (this parameter is kept fixed and equal to its optimal 4 value). Figure 4 shows the success probability p s, entangling power E p, and entangling efficiency E eff as functions of ϕ. Because of the high importance of the c-phase gate for quantum computation, we have tested its entangling power and efficiency also experimentally. To calculate the entangling power and efficiency from the experimental data we scanned the four-parametric space of all separable input states numerically (over uniformly distributed states were tested). The corresponding output states were calculated S-9

84 LEMR, ČERNOCH, SOUBUSTA, AND DUŠEK PHYSICAL REVIEW A 86, 0331 (01) p s N Np s 0 /8 /4 3 /8 / Angle 1 [rad] FIG. 5. C-phase gate. Success probability p s (full line), negativity N (dotted line), and their mutual product (dashed line) as functions of the parameter θ 1 for a fixed value of ϕ = π and θ = π 4. using the Choi matrices χ(ϕ), reconstructed by means of the quantum process tomography [18,0], using the following formula [1]: ρ out = Tr in [χ(ϕ)(ρin T 1)]. Entangling power and other quantities are shown in Fig. 4 where the values obtained from experimental data can be compared with the theoretical predictions. V. EXAMPLE 3: GENERALIZED C-PHASE GATE From the two previous examples the reader may get the impression that the entangling efficiency does not provide any substantial benefit with respect to entangling power. The reason lies in the specific nature of these examples. In the case of the beam splitter, negativity is independent on the input state, so it can be factored out of the maximum search in the formula for entangling efficiency [see Eq. (3)]. In the case of the c-phase gate the probability of success does not depend on the input state, so it can be taken out of the maximization. Therefore in both these situations the entangling power gives qualitatively the same results as the entangling efficiency gives. Here we expose the third example, the generalized c-phase gate, which proves that the entangling efficiency is, in general, a better instrument than the entangling power. The linear-optical scheme from Fig. 3 performs the c-phase gate transformation (10) only if a compensating filter, F1, with a proper transmissivity (γ = ps 1/4 ) is used in the upper path. If this filter is removed the device will perform the following generalized (nonunitary) transformation: 0,0 0,0, 0,1 4 ps (ϕ) 0,1, 1,0 4 ps (ϕ) 1,0, 1,1 p s (ϕ)e iϕ 1,1. (16) We will call it a generalized c-phase gate. Such a gate can be used, for instance, in quantum nondemolition measurement []. Quantity p s (ϕ) is defined by Eq. (11), but it does not represent the success probability of the generalized gate. The overall success probability P s is now a function of the input state P s (ϕ,θ 1,θ ) = cos θ 1 cos θ + p s (ϕ)(cos θ 1 sin θ + sin θ 1 cos θ ) + p s (ϕ)sin θ 1 sin θ. (17) max {Np s } max {N} P s N NP s 0 /1 /6 /4 /3 5 /1 / Angle 1 [rad] FIG. 6. Generalized c-phase gate. Success probability P s (full line), negativity N (dotted line), and their mutual product (dashed line) are plotted against the parameter of input state θ 1 for ϕ = π and θ = π/4. All the functions are symmetric in the sense that the same results would be obtained when exchanging θ 1 and θ. Assuming again pure product states parametrized by Eq. (1) in the input, a simple calculation reveals the formula for the negativity of corresponding output states N(ϕ,θ 1,θ ) = sin θ 1 sin θ ps (ϕ) 1 cos ϕ. (18) P s (ϕ,θ 1,θ ) It can be seen (look at Fig. 6) that the negativity alone is now maximized for generally different angles θ 1 and θ than in the case of the standard c-phase gate (Sec. IV). The product of the negativity and success probability, however, finds its maximum still in π/4. Figure 6 presents the success probability, negativity and their product as functions of θ 1 for ϕ = π and θ = π/4. Note that the relation is symmetric for θ. In this figure, one can clearly perceive that now negativity is maximized for different parameters than the product of negativity and success probability. Figure 7 is similarto Fig. 4 showing how entangling power and entangling efficiency varies with the phase shift ϕ. Because in the case of this gate the success probability is state dependent we plot here success probability P s (maxn) corresponding to the states which maximize the negativity. Besides, we have added function P s (maxn) E p. This function should help to view two P s (maxn) E p E eff E p P s (maxn) 0 /4 / 3 /4 Gate phase FIG. 7. Generalized c-phase gate. Entangling power E p (dotted line), corresponding success probability P s (maxn) (full line), their product (dotted-dashed grey line), and entangling efficiency E eff (dashed line) are plotted against gate phase shift ϕ S-10

85 ENTANGLING EFFICIENCY OF LINEAR-OPTICAL... PHYSICAL REVIEW A 86, 0331 (01) measures of the nonclassicality of the gate, entangling power and entangling efficiency, under comparable conditions. One can perceive that entangling efficiency is greater then the product P s (maxn) E p because it takes the success probability into the maximization. VI. CONCLUSION There is no doubt that a concept of a measure of the nonclassicality of quantum gates, which takes into account the success probability, is more natural for probabilistic devices than the concept of entangling power. The question was if such a measure, namely the entangling efficiency defined above in this paper, can really offer different and more appropriate information in the case of linear optical devices than the entangling power. Our last example shows that it can. In general, entangling efficiency is not a trivial function of entangling power. It indicates that entangling efficiency is a useful measure of the entangling capability of probabilistic quantum gates and that entangling power may sometimes yield deficient information about this capability. ACKNOWLEDGMENTS This work was supported by the Palacký University (PrF ), by the Institute of Physics of the Czech Academy of Sciences (AVOZ101005), and by the Czech Science Foundation (P05/1/038). [1]M.A.NielsenandI.L.Chuang,Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 000). [] L. M. K. Vandersypen and I. L. Chuang, Rev. Mod. Phys. 76, 1037 (005). [3] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (005). [4] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (007). [5] F. Schmidt-Kaler, H. Haffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature (London) 4, 408 (003). [6] N. Kiesel, C. Schmid, U. Weber, R. Ursin, and H. Weinfurter, Phys. Rev. Lett. 95, (005). [7] A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O Brien, Nature Commun., 4 (011). [8] N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, Rev. Mod. Phys. 83, 33 (011). [9] E. Knill, R. Laflamme, and G. J. Milburn, Nature (London) 409, 46 (001). [10] D. E. Browne and T. Rudolph, Phys.Rev.Lett.95, (005). [11] P. Zanardi, Ch. Zalka, and L. Faoro, Phys.Rev.A6, (R) (000). [1] M. M. Wolf, J. Eisert, and M. B. Plenio, Phys. Rev. Lett. 90, (003). [13] J. Batle, M. Casas, A. Plastino, and A. R. Plastino, Opt. Spectrosc. 99, 371 (005). [14] P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys.Rev.A60, 773R (1999). [15] V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998). [16] E. M. Rains, Phys. Rev. A 60, 173 (1999). [17] G. Vidal and R. F. Werner, Phys. Rev. A 65, (00). [18] K. Lemr, A. Černoch, J. Soubusta, K. Kieling, J. Eisert, and M. Dušek, Phys.Rev.Lett.106, (011). [19] K. Kieling, J. O Brien, and J. Eisert, New J. Phys. 1, (010). [0] J. F. Poyatos, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 78, 390 (1997). [1] M.-D. Choi, Linear Algebr. Appl. 10, 85 (1975). []G.J.Pryde,J.L.O Brien,A.G.White,S.D.Bartlett,andT.C. Ralph, Phys. Rev. Lett. 9, (004) S-11

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87 IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 44 (011) (5pp) doi: / /44/19/ Preparation of Knill Laflamme Milburn states using a tunable controlled phase gate Karel Lemr RCPTM, Joint Laboratory of Optics of Palacky University and Institute of Physics of Academy of Sciences of the Czech Republic, Faculty of Science, Palacky University, 17. listopadu 1, Olomouc, Czech Republic lemr@jointlab.upol.cz Received 3 May 011, in final form August 011 Published 30 August 011 Online at stacks.iop.org/jphysb/44/ Abstract A specific class of partially entangled states known as Knill Laflamme Milburn states (or KLM states) has been proved to be useful in relation to quantum information processing (Knill et al 001 Nature ). Although the usage of such states is widely investigated, considerably less effort has been invested into experimentally accessible preparation schemes. This paper discusses the possibility of employing a tunable controlled phase gate to generate an arbitrary KLM state. In the first part, the idea of using the controlled phase gate is explained on the case of two-qubit KLM states. Optimization of the proposed scheme is then discussed for the framework of linear optics. Subsequent generalization of the scheme to the arbitrary n-qubit KLM state is derived in the second part of this paper. 1. Introduction Important developments have been demonstrated in quantum information processing (QIP) in the past few decades [ 4]. Several outcomes of this scientific field such as quantum cryptography [5 8] or random number generation [9 13] have already found their industrial applications. In other cases, a lot of effort has yet to be invested into the research. Mainly the lack of some experimental tools (e.g. strong optical nonlinearity [14]) prevents the development of efficient quantum devices. An important discovery has been achieved by Knill, Laflamme and Milburn [1], when they derived that a specific class of partially entangled states (so-called Knill Laflamme Milburn states, or simply KLM states) can be used to significantly improve the efficiency of quantum computing. They proposed a nearly deterministic teleportation-based protocol for quantum computation using the KLM states as ancillas. In this protocol, the overall success probability of quantum computation goes asymptotically to unity with a growing number of photons in the ancillary KLM state. Their work has been followed by several other related proposals and experiments [15 17]. Franson et al [15] have generalized the original KLM scheme so that the success probability of quantum computing scales better with a growing number of photons, but at the expense of lower fidelity of the output states. Several schemes for the preparation of KLM states have also already been proposed. The general preparation idea has been mentioned in the original KLM paper [1], though there was no specific recipe. The first explicit scheme for the preparation of the KLM states was proposed by Franson et al and it uses non-deterministic controlled sign gates and single photon interference to generate arbitrary photonnumber KLM states [18]. Another scheme limited only to two-photon KLM states, but not requiring any post-selection, was also proposed [19] and subsequently experimentally implemented [0]. This paper investigates yet another approach for experimentally accessible preparation of KLM states using the controlled phase gate (c-phase gate). The advantage of using this gate is the fact that the c-phase gate is considered an important part of the QIP toolbox [1, ]. The Franson et al scheme also employs the controlled phase gates (or in their case controlled sign gates), but with a constant phase shift set to π. In this paper, a fully tunable controlled phase gate is considered and a scheme for its usage as a resource for KLM state generation is developed. By this strategy, the overall success probability of the KLM state preparation can be increased considerably for some KLM states, as is shown in /11/ $ IOP Publishing Ltd Printed in the UK & the USA S-13

88 J. Phys. B: At. Mol. Opt. Phys. 44 (011) KLemr Figure 1. Scheme of the proposed procedure for generation of two-qubit KLM states. The signal and control input qubit undergo a c-phase gate with a tunable phase shift ϕ yielding the two-qubit KLM state. this paper. The presented scheme is fully general and allows us to prepare KLM states of an arbitrary number of qubits. Also no previous entanglement between the input qubits is required as the entangling capability of the gate itself is sufficient. The fully tunable controlled phase gate capable of imposing any phase shift in the range from 0 to π has already been both proposed theoretically [3] and implemented experimentally [4] on the platform of linear optics and thus can be considered experimentally accessible.. Basic two-qubit scheme Using the qubit representation, one can express the n-qubit KLM state in the form of n ψ KLM = α j 1 j 0 n j. (1) j=0 The original definition by Knill, Laflamme and Milburn sets α j = 1 n+1 for j = 0,..., n, but the subsequent research carried out by Franson et al [15] indicates that additional benefits can be found in using general amplitudes α j. Their research revealed that one can increase the efficiency of teleporationbased quantum computing for instance by choosing triangleshaped amplitudes α j (that is, α 0 = α n = 0 and alpha linearly growing towards maximum at α n/ and then decreasing). This improvement is obtained at the expense of lower fidelity of the output state. (For more details, consult [15].) In the first part of this paper, let us consider the preparation of two-qubit KLM states (see figure 1). The generalization to an arbitrary number of qubits will be presented later. Using the general definition for the KLM states (1), one can find that the two-qubit KLM states are in the form of ψ two qubitklm = α α α 11, () where α j (for j = 0, 1, ) are arbitrary complex amplitudes following the normalization condition j=0 α j = 1. Having the target state well defined, let us now inspect the properties of the c-phase gate. The c-phase gate is a two-qubit quantum gate whose action in the gate s computational basis reads e iϕ 11 (3) with numbers in the brackets denoting the first and second qubit states. A general c-phase gate can be set to impose an arbitrary phase shift ϕ to the two-qubit state 11. Any signal and control qubit can be expressed in terms of the gate s computational basis ψ c,s =cos θ c,s 0 c,s +e iφc,s sin θ c,s 1 c,s, (4) where the indices c and s denote the control and signal qubit. Note that this state can always be prepared with high fidelity using only single-qubit transformations (e.g. wave plates in the case of photon polarization encoding). The separable input state ψ c ψ s is transformed by the gate yielding ψ OUT = cos θ c cos θ s 00 +e iφs cos θ c sin θ s 01 +e iφc sin θ c cos θ s 10 +e i(φc+φs+ϕ) sin θ c sin θ s 11. (5) Using the expression for signal qubit (4), the output state can be rewritten in the following form: ( ψ OUT = cos θ c 0ψ s +e iφc sin θ c τ 1ψs + ɛ 1ψs ), (6) where ψs is the orthogonal state to ψ s so that ψs ψ s =0 and the parameters τ and ɛ are defined as τ = ψ s (cos θ s 0 +e i(φs+ϕ) sin θ s 1 ) = cos θ s +e iϕ sin θ s, ɛ = ψs (cos θ s 0 +e i(φs+ϕ) sin θ s 1 ) = e iφs sin θ s cos θ s (1 e iϕ ). (7) After performing the single-qubit transformation ψ s 0, ψs 1 (8) in the signal mode, one can clearly recognize the two-qubit KLM state in the output state of the gate, ψ OUT = cos θ c 00 +e iφc τ sin θ c 10 +e iφc ɛ sin θ c 11. (9) The remaining task is to map the complex amplitudes in (9) to the original amplitudes α j and to show that any two-qubit KLM state is achievable. First let us consider the relative amplitude ratio and phase between α 0 and (α 1 + α ). Any amplitude ratio can easily be set just by the choice of the θ c parameter of the input control state, α 1 + α = tan θ α 0 c. (10) As for the phase, the freedom in setting any value of φ c assures that any phase shift between α 0 on one side and α 1 and α on the other side is achievable. The relation between α 1 and α is also simple. For instance, setting the phase shift ϕ = π simplifies the amplitude ratio to α α 1 = ɛ τ = tan θ s (11) and an arbitrary phase shift between α 1 and α can be set by the choice of φ s. Note that setting ϕ = π allows us to cover the whole class of KLM states. This fact will be used for the discussion in section 5. Equations (10) and (11) manifest that any amplitude ratio between α 0, α 1 and α is achievable since tan goes from 0 to. S-14

89 J. Phys. B: At. Mol. Opt. Phys. 44 (011) KLemr 3. Success probability optimization One may conclude that the tunability of the gate in the phase shift ϕ is a redundant feature. However, this parameter can be used for optimization of the procedure. One of the most promising platforms for QIP is linear optics [5 9]. For this reason, let us now focus on the optimization of the proposed procedure for linear optics. Recently, Kieling et al [3] have identified the maximum success probability of a c-phase gate in the framework of linear optics as ( P C (ϕ) = 1+ sin ϕ + 3/ sin π ϕ sin ϕ 1/), 4 (1) which does not depend on the input state. The optimization of the proposed scheme seeks to maximize the success probability of the c-phase gate used for KLM state preparation. With respect to that a numerical simulation (or optimization) has been carried out to reveal the maximum achievable success probability for several KLM states. The target KLM state of the presented numerical simulation is the mono-parametric class of the two-qubit KLM state motivated by Franson et al s definition [15] (triangle-shaped amplitude function) ψ KLM = α α α (13) The amplitudes α 0 and α 1 are now considered to be real numbers as it has been shown above that the phase can always be set by the choice of φ c and φ s. These phases are independent of the gate phase shift ϕ and therefore have no effect on the success probability. The presented optimization will focus on the amplitude ratio α 0 /α 1 and investigate the corresponding success probability. The first numerical simulation has been performed to determine the maximum achievable α 0 /α 1 ratio for a given phase shift. Results of this simulation are presented in figure. One can observe that the maximum achievable α 0 /α 1 ratio grows monotonically with the phase shift ϕ. For reference, the success probability (1) as a function of the phase shift ϕ is also depicted along with the reference ratio α 0 /α 1 =1corresponding to the original KLM state definition. The second numerical simulation has been carried out to determine the maximum achievable success probability for a given α 0 /α 1 ratio (see figure 3). Also the setting of the phase shift ϕ and the parameter of the signal qubit θ s are depicted to illustrate the optimal strategy. This strategy is different in two regions separated by the amplitude ratio α 0 /α In the first region ( α 0 /α ) setting θ s = π and the phase 4 shift ϕ accordingly is the optimal way. One tries to minimize the phase shift used for the KLM state preparation, because the success probability is a decreasing function of the phase shift. To keep the phase shift minimal, one has to set θ s = π 4, because for a given phase shift ϕ the setting θ s = π 4 maximizes the α 0 /α 1 ratio. On the other hand, in the second region ( α 0 /α 1 > 0.54) the previously mentioned strategy will not yield optimal results. This is because of the success probability not being monotonic in this region. Setting ϕ = π and adjusting θ s instead is the optimal way here. Figure. Maximum achievable α 0 /α 1 ratio for a given phase shift of the c-phase gate. The success probability of the optimal linear optical c-phase gate as a function of its phase shift is also depicted for reference. Figure 3. Maximum achievable success probability and the corresponding optimal θ s and ϕ parameters are plotted as a function of the α 0 /α 1 ratio. Note that the optimal setting of ϕ for α 0 /α 1 > 0.54 is ϕ = π (this explains the step of ϕ at α 0 /α 1 =0.54). Both this and the original Franson et al scheme require n 1 times using the c-phase gate in order to generate the n- qubit KLM state. This leads to the overall success probability for the n-qubit KLM state, n 1 P KLM = P C (ϕ i ), (14) i=1 where n denotes the number of qubits and P C (ϕ i ) is the success probability of the controlled phase gate set for the phase shift ϕ i used in the i th repetition of the c-phase gate. The Franson et al proposal considers only ϕ i = π for all values of i. So for example in the two-qubit case, the success probability of the Franson scheme would yield a constant value of 0.11 (based on the optimal linear optical controlled phase gate). To emphasize the improvement achieved by the tunability of the phase gate, let us consider an example of α 0 /α 1 =0.5. For this particular choice, the success probability of the scheme proposed in this paper would be 3 S-15

90 J. Phys. B: At. Mol. Opt. Phys. 44 (011) KLemr 5. Optimization of the generalized scheme Figure 4. Generalization of the two-qubit scheme to an arbitrary number of qubits. The input n-qubit KLM state is combined with a new qubit initially in the 0 state. H denotes the Hadamard gate and C denotes the c-phase gate (this time set to impose the phase shift ϕ = π ). 0.18, which is a 60% improvement. This improvement in success probability varies with the particular choice of the target KLM state (see figure 3). 4. Generalization to n-qubit KLM states In the second part of this paper, the proposed two-qubit scheme is generalized to prepare KLM states of an arbitrary number of qubits. For simplicity let us now presume all complex amplitudes of the n-qubit KLM state being equal (original KLM state definition). To illustrate the generalization procedure, the step from the two-qubit to three-qubit KLM state is explained and also illustrated in figure 4). Going from the two- to three-qubit KLM state means that the following transformation has to be performed: (15) (16) , (17) where the indices 1 and denote the first and second original qubits of the two-qubit KLM state and the index 3 denotes the newly added qubit. This transformation can be implemented by the addition of a new qubit initially in the state 0. This new qubit is first subjected to the Hadamard gate (18) After that it is propagated through the c-phase gate set to the phase ϕ = π along with the last of the original KLM qubits. At the end, an inverse Hadamard gate is placed in the new qubit mode. One can see that in the case of the last original qubit being 0, the phase shift imposed to the new qubit is zero and the new qubit leaves the scheme in the state 0. On the other hand, if the last original qubit is in the state 1, the new qubit gets a π phase shift and yields after leaving the inverse Hadamard gate. The generalization to an arbitrary number of qubits is straightforward. To generate an (n +1)-qubit KLM state from an n-qubit KLM state (n ), a new qubit is added at the end of the original qubits and subjected to the procedure described in the previous paragraph. The general scheme is depicted in figure 4. The previous section is just a proof of the scalability of the scheme but does not give optimal setting with respect to the success probability. A similar optimization as for the twoqubit KLM states can be considered to maximize the yield of the scheme. Hadamard gates can be replaced by more general single-qubit transformations and together with the tunability of the phase shift imposed by every controlled phase gate, the overall success probability can be optimized with respect to the selected target KLM state. One can use the iterative procedure starting from the n-qubit KLM state with amplitudes α [n] j, j = 0,..., n, and going to the (n +1)-qubit KLM state with amplitudes α [n+1] j, j = 0,..., n + 1. Here the upper index denotes the n-qubit starting KLM state and (n + 1)-qubit target KLM state. Note that in this case the c-phase gate is applied to the last of the original qubits (nth qubit) and a newly added (n +1)th qubit. This new qubit can be expressed in the form of ψ s as defined by (4) and the last original qubit takes effectively the form similar to ψ c with cos θ c = n 1 α [n] j (corresponding to the 0 state) j=0 sin θ c = α n [n] (corresponding to the 1 state) φ c = arg ( α n [n] ) (19) also following the original definition (4). With this mapping, one can proceed in a similar way as explicitly described in section. The resulting amplitudes α [n+1] j are then expressed in the form of α [n+1] j α n [n+1] = α [n] j, for j = 0,..., n 1 = α n [n] eiφc τ α [n+1] n+1 = α n [n] eiφc ɛ, (0) where φ c is defined by (19) and τ and ɛ by (7). The equations become increasingly complicated with the growing number of qubits. For this reason, one can seek the solution numerically. As a result of such a numerical optimization, one can for example prepare a four-qubit KLM state of the triangle-shaped amplitudes in the form of ψ KLM = 1 4 α j 1 j 0 n j (1) N j=0 α 0 = α 4 = 1, α 1 = α 3 = 3, α = 6 () n (N = j=0 α j ) with the success probability of 0.19%, while the original proposal would give only 0.14% success probability (40% improvement). Note that this improved success probability would allow almost 1.5 times higher rate of preparation of KLM states for the nearly deterministic protocol proposed by Knill, Laflamme and Milburn [1]. The reason for the improvement in the success probability is the fact that using a tunable phase shift, one can operate the controlled phase gate at an optimal phase shift. Because one can always set the gate to operate at the phase π and set single-qubit 4 S-16

91 J. Phys. B: At. Mol. Opt. Phys. 44 (011) KLemr operations accordingly, the proposed scheme would never give lower success probability as the one proposed by Franson et al. The optimal strategy for setting the phase shift imposed by the gate in every step of the generalized procedure is similar to the strategy discussed in section 3 for the two-qubit case. This can be summarized by an inequality n 1 P KLMFranson = P C (π) = P C (π) n 1 i=1 n 1 P KLMnew = P C (ϕ i ), (3) where the left-hand side corresponds to the success probability of the Franson et al proposal and the right-hand side corresponds to the success probability of the scheme described in this paper. In the worst-case scenario, the hereby proposed scheme allows us to set ϕ = π to generate any KLM state and in this case the inequality would be saturated. 6. Conclusions The scheme presented in this paper shows how a tunable controlled phase gate can be used to generate arbitrary n-qubit KLM states. In comparison with the Franson et al proposal, this scheme gives higher success probability depending on the requested KLM state. It can offer a significant improvement in generation of ancillary states for efficient quantum computing. Note that this paper discusses the improved generation success probability (rate) for the KLM ancillary states. It should not be confused with the success probability of the teleportationbased KLM scheme that employs these ancillary states and considers them as already prepared. Several specific KLM states are discussed in this paper and their preparation success probabilities shown to demonstrate this improvement. Acknowledgments The author would like to thank his colleagues Jan Soubusta and Antonín Černoch for fruitful discussion on the subject of this paper. The author gratefully acknowledges the support by the Operational Program Research and Development for Innovations, European Regional Development Fund (project CZ.1.05/.1.00/ ), and the Operational Program Education for Competitiveness, European Social Fund (project CZ.1.07/.3.00/0.0017), of the Ministry of Education, Youth and Sports of the Czech Republic, by Palacky University i=1 (internal grant PrF ). This paper is dedicated to my girlfriend Barborka. References [1] Knill E, Laflamme R and Milburn G J 001 Nature [] Alber G, Beth T and Horodecki M 001 Quantum Information (Berlin: Springer) [3] Bowmeester D, Ekert A and Zeilinger A 000 The Physics of Quantum Information (Berlin: Springer) [4] Nielsen M and Chuang I 00 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) [5] Benett C and Brassard G 1984 IEEE Int. Conf. on Computers, Systems, and Signal Processing p 175 [6] Benett C, Bessette F, Brassard G, Salvail L and Smolin J 199 J. Cryptol. 5 3 [7] Ekert A K 1991 Phys. Rev. Lett [8] Gisin N, Ribordy G, Tittel W and Zbinden H 00 Rev. Mod. Phys [9] Inoue H, Kumahora H, Yoshizawa Y, Ichimura M and Miyatake O 1983 Appl. Stat [10] Peres Y 199 Ann. Stat [11] Jennewein T, Achleitner U, Weihs G, Weinfurter H and Zeilinger A 000 Rev. Sci. Instrum [1] Stefanov A, Gisin N, Guinnard O, Guinnard L and Zbinden H 000 J. Mod. Opt [13] Katsoprinakis G E, Polis M, Tavernarakis A, Dellis A T and Kominis I K 008 Phys. Rev. A [14] Turchette Q A, Hood C J, Lange W, Mabuchiand H and Kimble H J 1995 Phys. Rev. Lett [15] Franson J D, Donegan M, Fitch M, Jacobs B and Pittman T 00 Phys. Rev. Lett [16] Okamoto R, O Brien J L, Hofmann H F and Takeuchi S 010 arxiv: v1 [17] Grudka A and Modlawska J 008 Phys. Rev. A [18] Franson J D, Donegan M M and Jacobs B C 004 Phys. Rev. A [19] Lemr K and Fiurášek J 008 Phys. Rev. A [0] Lemr K, Černoch A, Soubusta J and Fiurášek J 010 Phys. Rev. A [1] Sleator T and Weinfurter H 1995 Phys.Rev.Lett [] Barenco A, Bennett C H, Cleve R, DiVincenzo D P, Margolus N, Shor P, Sleator T, Smolin J A and Weinfurter H 1995 Phys. Rev. A [3] Kieling K, O Brien J and Eisert J 010 New J. Phys [4] Lemr K, ČernochA,SoubustaJ,KielingK,EisertJandDušek M 011 Phys. Rev. Lett [5] Munro W J, Nemoto K, Spiller T P, Barrett S D, Kok P and Beausoleil R G 005 J. Opt. B 7 S135 [6] O Brien J L 007 Science [7] Walmsley I A 008 Science [8] Aspelmeyer M and Eisert J 008 Nature [9] Politi A, Cryan M J, Rarity J G, Yu S Y and O Brien J L 008 Science S-17

92 S-18

93 Optics Communications 300 (013) 8 85 Contents lists available at SciVerse ScienceDirect Optics Communications journal homepage: Linear-optical programmable quantum router Karel Lemr a,n, Antonín Černoch b a RCPTM, Joint Laboratory of Optics of Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic, 17. listopadu 1, Olomouc, Czech Republic b Institute of Physics of Academy of Sciences of the Czech Republic, Joint Laboratory of Optics of PU and IP AS CR, 17. listopadu 50A, Olomouc, Czech Republic article info Article history: Received 1 November 01 Received in revised form 1 January 013 Accepted 3 February 013 Available online 16 March 013 Keywords: Quantum router Quantum information processing Photon pairs Quantum communications Programmable phase gate abstract This paper presents a scheme for linear-optical implementation of a programmable quantum router. Polarization encoded photon qubit is coherently routed to two spatial modes according to the state of control qubits. In our implementation, the polarization state of the signal photon does not change under the routing operation. We also discuss generalization of the scheme that would allow to obtain signal dependent routing. & 013 Elsevier B.V. All rights reserved. 1. Introduction Quantum information processing is a joint field of physics and information science [1]. It is dedicated to improve the capability to transmit and process information by employing laws of quantum physics. In many aspects, the quantum information processing is inspired by its classical analogue. For instance the proposed quantum networks mimic the geometry of their classical counterparts []. Also the elementary gates for both quantum and classical information processing devices are considerably alike [3]. An important building block of classical information networks are the so-called routers devices used to direct the information from the source to its intended destination [4]. The more complex the network is, the more pronounced is the need for correct routing of the information. Classical routers are indeed densely used and are subject to intensive applied research [5 7]. On the other hand, their quantum analogues the quantum routers have not yet been investigated so intensely. Quantum routing can be achieved for instance by employing matter light interaction [8 11]. These techniques might however prove experimentally highly demanding. An all-optical quantum router has also been presented in Ref. [1], but this device use only classical information to control the routing. Very recently, Chang et al. proposed a strategy for linear-optical quantum routing based on the entanglement between signal and control n Corresponding author. Tel.: þ address: k.lemr@upol.cz (K. Lemr). qubit [13]. In their proposal the routing control is represented by a qubit. However, the information stored in the signal qubit collapses depending on the measurement performed on the control qubit. Also the requirement to maintain entanglement between the routed and control qubit may be limiting in practical quantum networks. In this paper we propose a scheme for implementation of an all-linear-optical programmable quantum router (see conceptual scheme in Fig. 1). As the title suggests, the signal qubit is routed according to the state of control qubit(s). Since the device is quantum, it can route the signal qubit to a coherent superposition of output modes. Our implementation combines the benefits of the above mentioned implementations. It is all-optical, fully quantum (routing control is achieved by means of a qubit) and it does not require any previous entanglement between signal and control qubit. On top of that, the information stored in the signal qubit does not change under the routing procedure. Since the device is based solely on linear-optical components, it can be implemented experimentally with presently available technologies [14]. We employ two distinct types of quantum information encoding. Polarization encoding is used for storing the signal and control information while spatial mode encoding is then used for routing.. Principle of operation The linear-optical implementation of the quantum router is depicted in Fig.. As already stated above, the signal qubit is /$ - see front matter & 013 Elsevier B.V. All rights reserved. S-19

94 K. Lemr, A. Černoch / Optics Communications 300 (013) following unitary transformation on the signal state U PPG ðf 1 Þ9HS ¼ 9HS, U PPG ðf 1 Þ9VS ¼ e if 1 9VS, ð3þ Fig. 1. Conceptual scheme of a programmable quantum router. The input signal qubit 9C ss initially in spatial mode 1 is coherently routed to output modes 1 and depending on the state of the control qubit(s). Since the quantum information itself is stored in other then spatial degree of freedom (we use polarization encoding), it is not modified by the routing procedure. Fig.. Scheme of the linear-optical implementation of a programmable quantum router. PPG denotes the programmable phase gate, l= denotes half-wave plate and PBS stands for polarizing beam splitter (horizontal linear polarization is transmitted while vertical linear polarization is reflected). The signal input qubit is encoded into polarization state of the signal photon. After interacting with control qubits via the PPG, the signal photon is routed to a coherent superposition of the two output ports labelled 1 and. The polarization state of the signal photon is kept unchanged. stored in polarization state of the signal photon 9C s S 1 ¼ a9hsþb9vs, where 9HS denotes the horizontal and 9VS vertical linear polarization states and 9a9 þ9b9 ¼ 1. Index 1 expresses the fact that the signal qubit is initially in the first spatial mode. Similarly, the routing control is stored in polarization states of the control photons 9F c1 S and 9F c S 9F c1 S ¼ 9F c S ¼ p 1 ffiffiffi ð9hsþe if 1 9VSÞ: ðþ Note that both the control qubits are in the same state. The router works as follows: the signal is brought to the input port of the first polarizing beam splitter PBS 1. There the horizontal and vertical polarization components split and proceed by separate arms (horizontal polarization is transmitted while vertical is reflected). In these arms, both the polarization components undergo a programmable phase gate (PPG) [15] enveloped by two Hadamard gates placed in front and behind the PPG. The Hadamard gates are easily implemented on polarization encoded qubits just by using a half-wave plate rotated by.5 1with respect to horizontal polarization direction. The kernel of this router are the two PPGs where the signal qubit interacts with the control qubits. These two-photon gates can be implemented in linear optics by letting signal and control photons interfere on a polarizing beam splitter and subsequently performing projection to anti-diagonal and diagonal linear polarizations in the control output mode. Depending on this projection outcome, a feed-forward polarization correction is either applied or not to the signal photon. Assuming the control photons in the polarization state (), the PPG gates conditionally perform the ð1þ where f 1 is the phase shift as defined in Eq. (). The interaction between the signal and control qubits results in adding the phase shift f 1 to the signal qubit state. Since in one half of the cases, the signal and control photons bunch to the same output port, the success probability of the PPG is 1 and the successful events are postselected by observing exactly one photon in the control mode. The reader is encouraged to find out more about the PPG gates in [16,17]. An important property of the PPG gate is that the postselection on single photon in the control mode works as well in the case of the absence of the signal photon. In this case the gate preserves a vacuum state in the signal mode. Also from the outcome of the control photon projection there is no telling whether there was or was not the signal photon. This property of the PPGs assures that in the router the signal photon state does not collapse and remains in a coherent superposition between the upper and lower arm. One can easily verify that the above mentioned procedure renders the signal state to the form 9C s S 1 - ð1þeif 1 Þ ða9hs 1 þb9vs Þþ ð1 eif 1 Þ ða9vs 1 þb9hs Þ, ð4þ After that the signal is recombined on the second polarizing beam splitter PBS. 1 The entire signal transformation now reads 9C s S 1-1 ½ð1þeif 1 Þða9HS1 b9vs 1 Þþð1 e if 1 Þðb9HS þa9vs ÞŠ: ð5þ Simple corrections are needed in both output modes to revert the signal polarization state back to its original form. Namely in the first output mode, one needs to perform a p phase shift between horizontal and vertical polarizations. This is achieved by a halfwave plate rotated by 0 1. In the second output mode, the undesired polarization HV swap is compensated by another half-wave plate with optical axis rotated by Together with these corrections the entire routing operation can be expressed in the form 9C s S 1 -A 1 9C s S 1 þa 9C s S, ð6þ where A 1 ¼ 1 ð1þeif 1 Þ, A ¼ 1 ð1 eif 1 Þ: ð7þ There is a clear formal resemblance between the routing transformation and the action of a polarization independent beam splitter. One can explore this similarity by introducing transmissivity T and reflectivity R and describing the router in terms a programmable beam splitter p 9C s S 1 - ffiffiffi p T 9Cs S 1 þ ffiffiffi R e iðp=þ 9C s S, ð8þ where T ¼ 1 ð1þcos f 1 Þ, R ¼ 1 T ¼ 1 ð1 cos f 1Þ: ð9þ Note that by means of the control qubits 9F c1 S and 9F c S one can fully tune the intensity transmissivity T from 0 to 1. On the other hand, there is no control over the phase shift between two output modes which is fixed to a value of p. This limitation does not burden applications where the router is used just for simple signal directing. Since two PPGs are employed, the overall success probability of the router is We adhere to the beam splitter convention of p phase shift added to the term corresponding to reflection from second to first mode. Alternatively one can imagine a wave plate added to the second output port of the polarizing beam splitter shifting the vertical polarization by phase shift p with respect to the horizontal one. S-0

95 84 K. Lemr, A. Černoch / Optics Communications 300 (013) Conclusions This paper brings forward a proposal for experimentally feasible programmable quantum router based on linear optics. The device, like many other linear-optical QIP devices [18 1], Fig. 3. Additional block of two PPGs used to set required phase shift between the two output modes of the quantum router. Components are labelled as in Fig.. If there is also a need for programmable control of the phase shift, an additional apparatus has to be added to one of the output modes. Suppose it is added to the second output mode. This additional block of optical components (see Fig. 3) is composed of similar components as the original routing block. Polarizing beam splitters PBS 3 and PBS 4 forming an interferometer are used with two PPGs, one in each arm of this interferometer. This time, there are no Hadamard gates present. Instead of that, the arm that processes horizontal component of the signal state is equipped with two half-wave plates rotated by These wave plates encompass the PPG and swap the horizontal polarization to vertical and then back again. Direct calculation reveals that the action of this entire additional apparatus shifts the signal state in the second arm by the phase denoted f 3 arising from the definition of control qubit states 9F c3 S and 9F c4 S 9F c3 S ¼ 9F c4 S ¼ p 1 ffiffiffi ð9hsþe if 3 9VSÞ: ð10þ Including the additional block, the overall action of the router represents the transformation p 9C s S 1 - ffiffiffi p T 9Cs S 1 þ ffiffiffi R e iðf ðp=þþ 3 9C s S : ð11þ This way the router can control not only the intensity splitting of the signal, but also the phase shift between the output modes. The price to pay for this additional degree of freedom is the decreased success probability of the value of Generalized action The action of the router can be generalized to provide more complex signal dependent routing. For this task, the conditions previously imposed on the control qubit states 9F c1 S ¼ 9F c S, 9F c3 S ¼ 9F c4 S ð1þ are abandoned and independent control qubits in the form 9F cn S ¼ p 1 ffiffiffi ð9hsþe if N 9VSÞ, N ¼ 1,,3,4 ð13þ are used. The router then performs the following generalized transformation 9HS 1 -e if pffiffiffiffiffi p 1 = ð T 19HS1 þ ffiffiffiffiffi R 1e iðf ðp=þþ 3 9HS Þ 9VS 1 -e if pffiffiffiffiffi p = ð T 9VS1 þ ffiffiffiffiffi R e iðf ðp=þþ 4 9VS Þ, ð14þ where indexes behind brackets denote spatial modes as usual while T N and R N for N¼1, are functions of the control states 9F c1 S and 9F c S T N ¼ 1 ð1þcos f N Þ, R N ¼ 1 T N ¼ 1 ð1 cos f NÞ: ð15þ works probabilistically, specifically with success probability of resp. 16 if control over phase between the output modes is required. In principle, the device can be constructed using the elementary set of single and two-qubit gates []. However taking into account the maximum obtainable success probability of some of these elementary gates, the scheme presented in this paper provides a significantly higher efficiency. Our router is capable of routing a polarization signal qubit to two distinct spatial output modes depending on the state of the control qubits. In contrast to routers based on classical information routing, the hereby presented device can benefit from its full quantum nature. For instance the control qubits can be entangled with other qubits in a larger scheme or can be direct outcomes of another quantum information processing algorithm. This would allow straightforward incorporation of our router to future complex quantum information networks. Even though it requires 4 control qubits, it only uses twoqubit PPGs which has already been achieved experimentally [16,17]. The need for more then one control qubit lies in the fact that one needs to address the individual logical basis states of the signal qubit separately. Similar resource demand can also be found in [4], where more detailed analysis concerning resource optimality is presented. One can also attempt phase covariant cloning of one control qubit to two or even four clones. This can be achieved with average fidelity of 85.4% (for 1- cloning) and linear optics is sufficient for this task [5]. Choosing this approach, one would need only one input control qubit at the expense of lower fidelity of the routing transformation. Finally a generalization of the scheme is discussed showing how to modify the router to perform signal dependent routing, where the need for multiple control photons is also justifiable. Acknowledgment The authors thank Jan Soubusta for fruitful discussion on the subject of this paper. The authors also gratefully acknowledge the support by the Operational Program Research and Development for Innovations European Regional Development Fund (project CZ.1.05/.1.00/ and the Operational Program Education for Competitiveness European Social Fund (projects Nos. CZ.1.07/.3.00/0.0017, CZ.1.07/.3,00/0,0058, CZ.1.07/.3.00/ of the Ministry of Education, Youth and Sports of the Czech Republic, by the Institute of Physics of the Czech Academy of Sciences (AVOZ101005) and by the Czech Science Foundation (P05/1/038). References [1] M. Nielsen, I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 00. [] R. Van Meter, IEEE Network 6 (01) 59. [3] A. Barenco, D. Deutsch, A. Ekert, R. Jozsa, Physical Review Letters 74 (1995) [4] D. Medhi, K. Ramasamy, Network Routing: Algorithms, Protocols, and Architectures, Morgan Kaufmann, 007. [5] M.A. Duguay, J.W. Hansen, Applied Physics Letters 15 (1969) 19. [6] J.L. Jackel, E.M. Vogel, J.S. Aitchison, Applied Optics 9 (1990) 316. [7] J. Ruiqiang, et al., Optics Express 19 (011) 058. E.g. optimal linear-optical controlled-not gate without ancillary photons succeeds only in 1 9 cases [3]. Furthermore a router based on such elementary gates would need more than one controlled-not gate. S-1

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97 PHYSICAL REVIEW A 87, (013) Resource-efficient linear-optical quantum router Karel Lemr, 1,* Karol Bartkiewicz, 1 Antonín Černoch, 1 and Jan Soubusta 1 RCPTM, Joint Laboratory of Optics of Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic, 17 listopadu 1, Olomouc, Czech Republic Joint Laboratory of Optics of Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic, 17 listopadu 50A, Olomouc, Czech Republic (Received 18 April 013; published 5 June 013) An all-linear-optical scheme for a fully featured quantum router is presented. This device directs the signal photonic qubit according to the state of one control photonic qubit. In the introduction we formulate the list of requirements imposed on a fully quantum router. Then we describe our proposal, showing the exact principle of operation on a linear-optical scheme. Subsequently we provide a generalization of the scheme in order to optimize the success probability by means of a tunable controlled-phase gate. Finally, we show how one can modify the device to route multiple signal qubits using the same control qubit. DOI: /PhysRevA I. INTRODUCTION Quantum communications represent a very important part of a rapidly developing research area called quantum information processing [1,]. Communications networks are now an indispensable technology allowing people to transmit information quickly over large distances. In classical communications networks laws of classical physics are used to govern their operation. Recent research in quantum physics and quantum information suggests that quantum laws of nature can provide significant improvement of capabilities of communications devices [3 5], while using similar resources and keeping similar network architecture [6,7]. Not surprisingly, a significant amount of theoretical and experimental research has been dedicated to the concept of quantum communications networks [8]. The most notable result of this effort is a number of protocols for quantum cryptography [9,10], which is a method for unconditionally secure transmission of information using various quantum properties of information carriers. Quantum communications networks can benefit from purely quantum effects such as entanglement or the probabilistic nature of measurement. The more complex both the classical and quantum networks are, the more pronounced is the need for correct routing of the signal from its source to its intended destination [11,1]. Classical routers are well-known ingredients of classical networks allowing one to direct signal information according to control information (e.g., IP address) [13]. The analogy between classical and quantum networks suggests that complex quantum networks would also require elaborate routing protocols. This need is even more pronounced since in contrast to classical information one cannot perfectly duplicate an unknown qubit of quantum information [14]. However, approximate cloning is possible and has been intensely studied both theoretically and experimentally over the last decades [15 17], resulting in, e.g., establishing and implementing optimal state-dependent cloning for a wide class of qubit distributions [18,19]. Nevertheless the impossibility of perfect cloning prevents using the concept of multidirectional * k.lemr@upol.cz PACS number(s): Hk, 4.50.Dv, Lx broadcast (known from classical networks) in a quantum network. In this paper we address the problem of designing a quantum router (see the conceptional scheme in Fig. 1). We consider the platform of individual photons and linear optics because of its experimental accessibility and also because of the particular suitability of light for information transmission [0]. The router has to fulfill five requirements to be suitable for quantum communications networks: (1) Both the signal and control information have to be stored in quantum objects (qubits), and therefore routers using classical information to route the quantum signal are considered only semiquantum routers. () The signal information is unchanged under the routing operation; the degree of freedom used to store the signal qubit information has to be kept undisturbed. (3) The router has to be able to route the signal into a coherent superposition of both output modes. (4) The router has to work without any need for postselection on the signal output. If the router is probabilistic, successful operation can be identified by detection on the control state. (5) To optimize the resources of the quantum network, only an individual control qubit is required to direct one signal qubit. There have been several schemes for quantum routers already proposed; some of them have been experimentally implemented, but none of them meets all five requirements defined in the above paragraph. The first group of these proposals uses light-matter interaction in order to achieve quantum routing [1 3]. Such interaction is, however, often very challenging for experimental implementation. The second group of the proposals considers solely the platform of optical interactions to accomplish the routing. There are some proposals for a semiquantum router, where the control information is classical using intensive light pulses [4]. There are also proposals in which the control information is quantum, but the signal state collapses in the router so requirement is not met [5]. Recently we have proposed a fully quantum router using only linear optics [6]. This device, however, does not meet requirement 5 since it requires two quantum bits to control routing of one single qubit of signal information /013/87(6)/06333(7) American Physical Society S-3

98 LEMR, BARTKIEWICZ, ČERNOCH, AND SOUBUSTA PHYSICAL REVIEW A 87, (013) signal input Ψ s control input quantum router Ψ c =cosθ 0 +e ıϑ sin θ 1 signal output 1 A } 1(θ, ϑ) Ψ s 1 +A (θ, ϑ) Ψ s FIG. 1. (Color online) Conceptual scheme of a quantum router. II. PRINCIPLE OF OPERATION In this section we describe an all-linear-optical quantum router that meets all five requirements mentioned in the introduction. The router makes use of three quantum gates: controlled-phase gate (c-phase gate) [8 3], quantum nondemolition presence detection gate (QND gate) [33], and programmable-phase gate (PPG) [34 36]. In order for the setup to work completely without the need for signal postselection, both the QND and the detector D have to be equipped with photon-number resolving detectors. Without them, the router still works but does not fulfill requirement 4. Linear-optical schemes of all of these gates are already published and with the exception of the QND gate are also already tested experimentally. The reader is encouraged to get more details in the cited papers. Figure shows the scheme for linear-optical implementation of the quantum router. We propose using two degrees of freedom of individual photons: (1) polarization encoding, used to store both the signal and control qubits, and () path (spatial mode) encoding, used NDF Ψ s s IN signal input PBS 1 1 generalized c-phase gate: PDBS T = 1 λ/ PPG QND generalized c-phase gate control input PBS c Φ c PPG: D/A λ/ s 1 PBS c A 1 Ψ s 1 signal outputs A Ψ s s λ/ QND: PBS (4x) a all λ/ λ/ λ/ λ/ @45 D/A FIG.. (Color online) Scheme of the linear-optical implementation of a quantum router. PBS denotes polarizing beam splitter, PDBS denotes polarization dependent beam splitter, λ/ and λ/4 denote half- and quarter-wave plates, NDF denotes neutral-density filter, c-phase denotes controlled phase gate, QND denotes quantum nondemolition detector, PPG denotes programmable-phase gate, and D/A denotes polarization analysis (for more details see Ref. [7]). Red lines depict signal modes, while blue lines depict control modes. a 1 D/A for the routing operation; the signal photon is routed into superposition of two output ports. The signal qubit enters the setup using the input port s in, while the control qubit enters the router using port c. Let us assume the signal qubit takes the form of a general quantum polarization state: s =α H +β V, (1) where H and V denote the states of horizontal and vertical linear polarizations and α + β =1. For reasons apparent later, it is suitable to parametrize the state of the control qubit by angles θ and ϑ: c =cos θ H +e ıϑ sin θ V. () After the signal qubit enters the router, it is subjected to the polarizing beam splitter (PBS 1 ) transmitting horizontally polarized light and reflecting light of vertical polarization. At this point the state of both signal and control qubits reads s c1 =αcos θ H 1 H c +αe ıϑ sin θ H 1 V c + β cos θ V H c +βe ıϑ sin θ V V c, (3) where indices denote spatial modes of the qubits. For better readability, let us now consider the evolution of both arms of the interferometer (labeled in Fig. as 1 and ) separately starting with the lower arm 1 corresponding to the first two terms in Eq. (3): The horizontally polarized signal photon undergoes a generalized Hadamard transformation on a half-wave plate (HWP) rotated by 30 1 : H H 1 V 1. Subsequently both the signal and control qubit photons enter the c-phase gate implemented using polarization dependent beam splitter (PDBS) of intensity transmissivities T V = 1 3 and T H = 1 for vertical and horizontal polarization, respectively. Postselecting only on the cases in which there is one photon in signal and one photon in control mode, the transformation performed by the c-phase gate renders the investigated first two terms of Eq. (3) to α cos θ ( H 1 + V 1 ) H c + αeıϑ sin θ 3 ( H 1 V 1 ) V c. In the next step, the signal qubit undergoes Hadamard transform also using a HWP yielding the terms α cos θ H 1 H c + αeıϑ sin θ V 1 V c. 6 As mentioned above, the c-phase gate is successful only when the signal and control qubits leave the gate by separate output ports. In order to postselect only on such cases, the control qubit has to be subjected to QND presence detection via the QND gate. The QND presence detection requires an entangled pair of photons as a resource, but these photons are of a fixed quantum state, are generated locally solely for the purposes of 1 In the entire paper, rotations of wave plates are given as the angle between the optical axis of the wave plate and the direction of horizontal linear polarization S-4

99 RESOURCE-EFFICIENT LINEAR-OPTICAL QUANTUM ROUTER PHYSICAL REVIEW A 87, (013) the QND gate, and do not take another part in the quantum network. The control qubit state does not change under the QND gate, but a success probability factor of 1 is added to take into account the success probability of the QND gate. More information about the QND gate can be found in Ref. [33]. In the last step, the control qubit is subjected to a half-wave plate rotated at.5 yielding the terms α cos θ H 1 ( H c + V c ) + αeıϑ sin θ V 1 ( H c V c ), 6 and then the control qubit alone impinges on the PPG composed only of a PBS. The PPG gate heralds successful operation only if there is a control qubit with horizontal polarization at its input, thus projecting the signal in the lower arm 1 onto α cos θ H 1 + αeıϑ sin θ V 1, (4) 6 and in this form it impinges on the PBS. Now we examine the evolution of the second two terms in Eq. (3) corresponding to the propagation of the signal qubit by the upper arm : In this case the control qubit enters the c-phase gate alone, rendering the investigated terms to β cos θ V H c + βeıϑ sin θ V V c. 3 Again, we postselect only on those cases in which the control qubit leaves the c-phase gate by the mode c. This postselection is again assured by the QND gate, which witnesses the control qubit presence with a success probability of 1. Considering also the action of HWP we get β cos θ V ( H c + V c ) + βeıϑ sin θ V ( H c V c ). 3 Subsequently the signal qubit undergoes a Hadamard transformation yielding β cos θ ( H H c + H V c V H c V V c ) + βeıϑ sin θ ( H H c H V c V H c + V V c ). 6 In the next step, both the signal and control qubits meet on the PPG gate s polarizing beam splitter. Since we postselect only on the cases in which there is exactly one photon at the output of mode c, we continue considering only the terms β cos θ ( H H c V V c ) + βeıϑ sin θ ( H H c + V V c ). 6 As usual, the control photon impinges on the detector. Depending on the outcome of the polarizing detection measurement performed, the signal qubit collapses into β cos θ 4 ( H V ) + βeıϑ sin θ 4 ( H + V ), (5) 3 when the diagonally polarized control qubit was detected, or β cos θ 4 ( H + V ) + βeıϑ sin θ 4 ( H V ), 3 when we observe the antidiagonally polarized control photon. In the latter case, we do apply a feedforward consisting of a HWP placed at 0 causing V V and thus correcting the signal to the form of Eq. (5). The subsequent Hadamard gate renders the signal to the form of βe ıϑ sin θ H + β cos θ V. 3 Before the signal impinges on the second polarizing beam splitter PBS, we introduce filtering by a neutral-density filter (NDF) of transmissivity T = 1 to balance the amplitude with respect to the lower arm contribution described in Eq. (4). After having both the terms of Eq. (3) evaluated, let us recall that the total state of the signal qubit reads s = α cos θ H 1 + αeıϑ sin θ V βeıϑ sin θ H + β cos θ 6 V, and after being subjected to the PBS it takes the form of s = cos θ (α H 1 β V 1 ) + eıϑ sin θ 6 (β H +α V ). The additional half-wave plate at 0 in the first output corrects V +V, and polarization swap H V is applied to the second output by inserting a half-wave plate at 45 yielding the final form of the signal state at the output of the router: s out = A 1 s 1 + A s, (6) where one can clearly observe the routing operation. The polarization state remains in the original form of Eq. (1), but the spatial degree of freedom is modified depending on the parameter θ of the control qubit. The ratio between the amplitudes A and A 1 depends straightforwardly on this parameter: tan χ = A A 1 = tan θ, (7) 3 where we have introduced the routing ratio parameter χ in a similar manner as splitting ratio parametrization is introduced for ordinary beam splitters. Since θ lies in the interval [0; π ], the router can direct all the signal to the first output (A 1 ), to the second output (A ), or to any of their superposition. The success probability P succ = A 1 + A does not depend on the signal state parameters α and β but only on the ratio χ and therefore on the control qubit parameter θ.one can easily find the relation between success probability and the routing parameter χ or the control qubit parameter θ: P succ = 1 + cos θ = 1 + tan χ tan χ. (8) It reaches a maximum of 1 for χ = θ = 0, and, on the other 8 hand, it is minimized for χ = θ = π 1 to a value of. The plot 4 in Fig. 3 shows the success probability of routing as a function of the routing parameter χ. It also depicts the relation between the routing parameter χ and the control qubit parameter θ. Note that the neutral-density filter of the intensity transmissivity of 1 3 can be placed to output port s 1 in order to S-5

100 LEMR, BARTKIEWICZ, ČERNOCH, AND SOUBUSTA PHYSICAL REVIEW A 87, (013) Success probability P succ P succ /8 /4 3 /8 / [rad] / 3 /8 /4 /8 [rad] FIG. 3. (Color online) Success probability of the routing procedure as a function of the routing ratio parameter χ. Additionally the plot shows the relation between the routing parameter χ and the control qubit parameter θ. equalize the success probability so it is completely control state independent and fixed to the value of 1 4. III. TUNABLE C-PHASE GATE BASED ROUTER In the previous section we have considered only the c-phase gate with a fixed phase shift of ϕ = π (also known as the controlled-sign gate). In some cases, however, higher success probability can be achieved using a tunable c-phase gate that can be set to exercise a phase shift ϕ of any value in the range [0; π]. The reason for considering this tunable c-phase gate lies in the fact that the success probability of the gate is a function of its phase shift. In 010, Konrad Kieling and his colleagues [30] discovered the success probability P C relation to the phase shift ϕ : [ P C = 1 + sin ϕ ( ) π ϕ + 3/ sin sin ϕ 4 1/]. (9) In order to make the c-phase gate tunable, one needs to replace the fixed polarization dependent beam splitter by an interaction Mach-Zehnder interferometer with tunable phase and losses [see Fig. 4(a)]. In a recent paper [3], we showed that if one does not seek this success probability to be input state independent the c-phase gate can be generalized to perform the transformation H 1 H c H 1 H c, H 1 V c A C H 1 V c, V 1 H c A C V 1 H c, V 1 V c A C e ıϕ V 1 V c, V H c V H c, V V c A C V V c, (10) where we have already adopted the notation of Eq. (3) and introduced A C = P C. Note that in the first and fifth cases there is no photon entering the interaction interferometer, while in the second, third, and sixth cases there is exactly one photon entering this interferometer. In the fourth case, both the photons are subjected to the interferometer. The success probability thus becomes input state dependent but higher in average. P C is the optimal success probability achievable using only linear optics and vacuum ancillae. When using the tunable c-phase gate for routing, several modifications of the setup have to be put in place [see Fig. 4(b)]: First, the PPG gate has to be replaced by a second c-phase gate set to the same phase shift as the first one. This c-phase gate would exercise the same transformation as described by Eq. (10), but with indices 1 swapped. Second, the transformation (HWP) performed on the control qubit between the two interaction gates is removed, and so is the NDF in upper arm. Finally, a generalized transformation in both arms 1 and before the signal enters the c-phase gates has to be recalculated. This transformation reflecting the success probability of the c-phase gate reads in 1 and H 1 V AC 1 + AC H AC V 1 AC 1 H V 1 + AC 1 + AC in. This transformation assures the signal state independence of the routing procedure. After incorporating these modifications, one can proceed in exactly the same manner as in Sec. II to reveal that the signal state just before impinging on the PBS takes the form of α A C [ cos θ + AC e ıϑ sin θ(1 + e ıϕ ) ] H 1 + A C + αeıϑ sin θ A 3 C (1 e ıϕ ) V 1 (11) + A C in the first arm 1 and the form of β A C [ cos θ + AC e ıϑ sin θ(1 + e ıϕ ) ] V + A C + βeıϑ sin θ A 3 C (1 e ıϕ ) H (1) + A C in the second arm. TheEqs.(11) and (1) indicate that the value of the phase shift ϕ imposed by the gate limits the routing ratio χ: A C sin θ 1 e ıϕ tan χ = cos θ + A C e ıϑ sin θ(1 + e ıϕ ). For the sake of readability, we limit to ϑ = 0intheremaining part of this section. Thus, even for θ = π the routing ratio is bound by the relation ( 1 cos ϕ χ L (ϕ) max{χ(ϕ,θ)} θ = atan sin ϕ ) = ϕ, (13) where we have introduced the routing ratio limit χ L for a given phase shift ϕ. Using this definition, we can formulate the range of achievable routing ratios for a given phase shift ϕ to be χ [0; χ L ] obtained when tuning monotonically the control qubit in the range θ [0; π ]. Note that, as expected, χ L = 0 for ϕ = 0 and χ L = π for ϕ = π. We can also easily find the success probability of the router by calculating the norm of the output state Eqs. (11) and (1) S-6

101 RESOURCE-EFFICIENT LINEAR-OPTICAL QUANTUM ROUTER PHYSICAL REVIEW A 87, (013) (a) Φ c PBS PBS 1 Min. P succ Max. P succ H s (b) BS PBS PS BS NDF PBS D/A λ/ s 1 Success probability P succ /8 1/4 Ψ s PBS 1 c-phase λ/ λ/ PBS QND c-phase λ/ λ/ Φ c FIG. 4. (Color online) (a) Linear-optical scheme of a tunable c-phase gate. The interaction Mach-Zehnder interferometer replaces the fixed polarization dependent beam splitter (PDBS) in the original scheme. The phase shift imposed by the gate is tuned by setting specific values of phase shift (PS) and losses (NDF) in this interferometer (for more details see Ref. [31]). (b) Modified scheme of the router using two tunable c-phase gates. Red lines depict signal modes, while blue lines depict control modes. This success probability function depends on both ϕ and θ. For a fixed value of ϕ we can find its minimum of A 4 min{p succ (ϕ,θ)} θ = 3 C (14) + A C always for θ = π. On the other hand, the maximum success probability is obtained for different values of θ depending on the phase shift ϕ: θ Pmax = 1 atan [ AC (1 cos ϕ) 1 A C λ/ s ]. (15) Both the success probability and the routing ratio limit are functions of the gate phase shift ϕ. To illustrate their mutual relation, we have plotted the success probability as a function routing limit in Fig. 5. In this figure, we show the maximum and minimum success probability together with the range between them for a given routing ratio limit. Observing this plot, we conclude that for sufficiently small values of routing ratio limit (e.g., χ L = π ) the tunable c-phase gate offers increased 8 success probability in comparison with the scheme proposed in the previous section. On the other hand, for larger routing ratio limits (closer to π ), the former scheme does better since it uses only one c-phase gate together with a more efficient PPG gate. IV. MULTIQUBIT ROUTING So far we have only considered routing a single signal qubit using one control qubit. In principle, however, the device can route a chain of signal qubits making use of the same /8 /4 3 /8 / Routing ratio limit L FIG. 5. (Color online) Maximum and minimum success probability of the router using two tunable c-phase gates depicted as functions of the routing ratio limit. The plot indicates that for sufficiently small values of routing ratio limit the tunable c-phase gate offers better performance than the scheme devised in the previous section (maximum and minimum success probability of the previous scheme is depicted using black dashed lines at 1 and 1 ). The dotted line shows 8 4 the success probability if the standard state-independent c-phase gate is used instead of the generalized version. control qubit. Figure 6 depicts the configuration for such multiqubit routing. In this model, two c-phase gates are used as in the previous section, but the control qubit is not detected immediately. After interacting with the first signal qubit, the control qubit presence is verified in a QND gate and then it is transferred back to the input of the router so it meets with the second signal qubit. After interacting with the last signal qubit, the control qubit is subjected to polarization analysis, thus projecting the state of all signal qubits. Let us assume the state of the nth signal qubit to be n =α n H +β n V. Thus, the state of the entire system can be expressed as c n in = (cos θ H +e ıϑ sin θ V ) n in, n n where the control qubit c has been introduced as in previous sections. The procedure described in the previous paragraph D/A QND } signal control c c router FIG. 6. (Color online) Scheme for multiqubit routing; c denotes the classical optical coupler. Red lines depict signal modes, while blue lines depict control modes. s 1 s S-7

102 LEMR, BARTKIEWICZ, ČERNOCH, AND SOUBUSTA PHYSICAL REVIEW A 87, (013) then renders the state to cos θ H n out1 + eıϑ sin θ V n out, 3 n n where out1 and out denote output ports of the router. Subsequent detection of the control qubit in the H ± V basis projects the state to the required form of cos θ n out1 + eıϑ sin θ n out. 3 n n Since the routing operation is probabilistic, the success probability of routing n qubits decreases exponentially with n. Considering the success probability of one run of the router to be P succ (as calculated in previous section), the n-qubit router will perform with success probability P total = 1 4n (1 8 9 sin θ) n. It is worth noting that, apart from routing, this procedure can be used to generate the so-called noon states [37,38]. These states of the form of ( N0 + 0N ), where N denotes the number of photons, are useful for instance in quantum metrology [39] or quantum lithography [40]. V. CONCLUSIONS We have presented an all-linear-optical scheme for a fully quantum router. The router meets all the requirements presented in the introduction. This concept is more practical than some of the previously presented routers making use of experimentally difficult light-matter interaction or unrealistic strengths of nonlinear optical phenomena [1 4]. Incontrast to previously published linear-optical schemes, our router provides genuine quantum routing being controlled by a qubit while not disturbing the state of the signal qubit (in contrast to [5]). Also, it only requires one control qubit to route a single signal qubit, making it more resource efficient (in contrast to [6]). The proposed scheme operates with success probabilities ranging between 1 8 and 1 depending on the control qubit state. 4 We also discuss optimization of the success probability that can be reached by using tunable c-phase gates. The presented analysis shows how efficiency can be increased up to 1 if only a small amplitude of the signal is needed to be sent to output port. Note also that detector efficiency just scales the success probability but does not change fidelity of the output. We have also provided a recipe for a multiqubit router, in which the same control qubit is used to route the general number of signal qubits. In this case, the success probability of the procedure scales exponentially with the number of routed signal qubits. While the entire scheme presented in this paper relies only on linear-optical quantum gates, it should be pointed out that the same setup can be constructed using gates based on nonlinear optical phenomena. Namely, the modified version as depicted in Fig. 4(b) can be transposed to nonlinear optics straightforwardly simply by replacing linear-optical c-phase gates by their nonlinear analogs. The benefit of such construction would be a significant improvement in success probability, since nonlinear c-phase gates can work deterministically. Also, the need for QND presence detection of the control qubit would become redundant. The nonlinear implementation of the c-phase gate is a heavily investigated topic currently facing severe experimental challenges [41 45]. ACKNOWLEDGMENTS The authors gratefully acknowledge support by the Operational Program Research and Development for Innovations, European Regional Development Fund (Grant No. CZ.1.05/.1.00/ ). K.L. acknowledges support by the Czech Science Foundation (Grant No P), K.B. acknowledges support by the Czech Ministry of Education (Grant No. CZ.1.07/.3.00/ ) and by the Polish National Science Centre (Grant No. DEC-011/03/B/ST/01903), A.Č. acknowledges support by the Czech Ministry of Education (Grant No. CZ.1.07/.3.00/0.0017), and J.S. acknowledges support by the Institute of Physics, CAS (Grant No. AVOZ101005). 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105 PHYSICAL REVIEW A 90, 0335 (014) Using quantum routers to implement quantum message authentication and Bell-state manipulation Karol Bartkiewicz, 1,,* Antonín Černoch, 3, and Karel Lemr, 1 Faculty of Physics, Adam Mickiewicz University, PL Poznań, Poland RCPTM, Joint Laboratory of Optics of Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic, 17. listopadu 1, Olomouc, Czech Republic 3 Institute of Physics of Academy of Sciences of the Czech Republic, Joint Laboratory of Optics of PU and IP AS CR, 17. listopadu 50A, Olomouc, Czech Republic (Received February 014; revised manuscript received 8 July 014; published 9 August 014) In this paper we investigate the capability of quantum routing (quantum state fusion) to implement two useful quantum communications protocols. The analyzed protocols include quantum authentication of quantum messages and nondestructive linear-optical Bell-state manipulation. We also present the concept of quantum decoupler a device implementing an inverse operation to quantum routing. We demonstrate that both quantum router and decoupler can work as specialized disentangling gates. DOI: /PhysRevA PACS number(s): Hk, Dd, 4.50.Dv, Lx I. INTRODUCTION Development of future quantum communications networks relies on successful implementation of several key protocols [1 3]. These protocols are quantum analogs of their classical counterparts such as amplification [4 10], secure key distribution [11 16], or error correction [17 3]. Another active element needed for the construction of complex quantum networks is quantum router a quantum-mechanical counterpart of the classical router used to steer the information from its source to intended destination. This component has been the subject of both theoretical and experimental research. In some cases, classical information is used to control the path of quantum signal [4], however other implementations can be called fully quantum routers since both the control and signal are quantum. In our recent papers, we have presented linear-optical schemes of quantum routers together with several criteria the router has to fulfill in order to be fully functional [5,6]. A similar quantum information protocol entitled quantum state fusion has been experimentally implemented by Vitelli et al. [7] independently of the above-mentioned studies. Other physical systems, e.g., light-atom interaction [8 33], have also been considered for construction of quantum routers. However, the quantum router has so far only been considered a mere replacement of the classical router for the purposes of quantum networks. In this paper, we go beyond this analogy and discuss the idea of using the quantum router to implement other useful quantum communications protocols. The quantum router will play a key role in complex quantum communications networks. Therefore, investigating its potential applications beyond the simple routing is of great significance. The article is organized as follows. In Sec. II, wereview the conceptual scheme of a quantum router and present its mathematical description. We also present the concept of a quantum decoupler, which is an inverse component to the quantum router. Specific linear-optical implementations of * bartkiewicz@jointlab.upol.cz cernoch@jointlab.upol.cz k.lemr@upol.cz both the router and the decoupler can be found in Refs. [5 7]. The following Sec. III describes a proposal for quantum authentication of quantum messages using the quantum router and decoupler. Finally, in Sec. IV we present a protocol for nondestructive Bell-state manipulation using the quantum router. We conclude in Sec. V. II. QUANTUM ROUTING AND DECOUPLING A. Quantum routing Quantum routing (or quantum state fusion [7]) is an essential protocol allowing for construction of complex quantum networks. In contrast to a classical router, the quantum router exploits quantum phenomena. Thus for instance the multidirection broadcasting (sending multiple copies of the signal to all output ports) known from classical routing is forbidden since it would contradict the no-cloning theorem. On the other hand, various quantum phenomena such as quantum superposition or entanglement can be used. In our recent paper, we have defined a set of requirements imposed on a fully functional quantum router [6]. These requirements include capability of the router to split quantum signal depending on the instructions passed by the control qubit while keeping the signal qubit intact. The basic one-to-two port quantum router is schematically depicted in Fig. 1. Assuming an initial signal qubit encoded in state ψ s and control qubit reads ψ c =γ 0 +δ 1. (1) The quantum router transforms the signal state ψ s to ψ s OUT = γ ψ s 1 + δ ψ s, () where indices 1 and denote output signal modes. Note that the signal state remains intact. Let us consider single photon qubits in polarization and spatial mode encoding. In this case, the signal qubit is encoded in polarization degree degree of freedom as ψ s =α H +β V, (3) where H and V denote horizontal and vertical polarizations, respectively. The control qubit is stored in another polarization /014/90()/0335(9) American Physical Society S-31

106 BARTKIEWICZ, ČERNOCH, AND LEMR PHYSICAL REVIEW A 90, 0335 (014) FIG. 3. (Color online) Scheme for polarization spatial encoding swap: PBS, fully polarizing beam splitter (transmitting horizontal polarization and reflecting vertial polarization); HWP, half-wave plate (rotated by 45 with respect to horizontal polarization). FIG. 1. (Color online) Conceptual scheme of a quantum router as described in the text. Signal qubit ψ s is routed into a coherent superposition of two output modes depending on the state of the control qubit ψ c. encoded photon state ψ c =γ H +δ V. (4) At the output, we find the signal photon leaving in superposition of both output ports depending on the control qubit ψ c while keeping the original signal qubit ψ s undisturbed. The state of the output is given as ψ s OUT = γ (α H +β V ) 1 + δ(α H +β V ), where indices 1 and denote spacial modes. B. Quantum decoupling One can also investigate inverse procedure to the routing transformation or quantum synthesis. In this case, two separate qubits are decoupled from an initial state expressed by Eq. (). FIG.. (Color online) Conceptual scheme of a quantum decoupler a device inverse to a quantum router. There are two qubits stored in the signal photon. The first qubit is encoded in its spatial degree of freedom and the second in polarization. Quantum decoupler facilitates interaction between this signal input state and an ancillary qubit leading to transcription of the spatial mode encoded qubit to the state of the ancillary qubit while the second signal qubit remains stored in the signal state. We will refer to this procedure as to quantum decoupling or quantum decomposition. The transformation implemented by the decoupler reads ψ s IN = γ ψ s 1 + δ ψ s ψ s ψ a, (5) where ψ a is the state of ancillary qubit ψ a =γ 0 +δ 1. The conceptual scheme of such a decoupler is shown in Fig.. In the analyzed case of linear-optical implementation with polarization and spatial mode encoding we express the decoupler transformation in terms of horizontal and vertical polarizations and spatial modes, ψ s IN = γ (α H +β V ) 1 + δ(α H +β V ) (α H +β V ) S (γ H +δ V ) A, (6) where S and A stand for signal and ancillary modes, respectively. C. Polarization spatial mode encoding swap Some applications (for instance the quantum authentication protocol described in Sec. III) require the capability to swap spatially and polarization encoded qubits before the signal is processed by the quantum decoupler. This task can easily be achieved deterministically using the setup depicted in Fig. 3. It consists of two half-wave plates rotated by 45 with respect to the horizontal polarization, and one fully polarizing beam splitter transmitting horizontal polarization and reflecting vertical polarization. In terms of signal state transformation, this component transforms ψ s =γ(α H +β V ) 1 + δ(α H +β V ) (7) into ψ s = α(γ H +δ V ) 1 + β(γ H +δ V ). (8) This operation when used for swapping the encoding of the signal input qubits before the decoupler is equivalent to swapping the signal and ancilla outputs of the decoupler. III. QUANTUM AUTHENTICATION OF A QUANTUM MESSAGE In both classical and quantum information research, the problem of message authentication is a heavily investigated topic [34]. The goal of all these protocols is to provide means to verify the authenticity of message origins and thus protect message from both the falsification and repudiation S-3

107 USING QUANTUM ROUTERS TO IMPLEMENT QUANTUM... PHYSICAL REVIEW A 90, 0335 (014) In 001, Gottesman and Chuang patented a scheme that used quantum information to securely sign classical messages [35,36]. In another seminal paper [37] published one year later, the authors have presented a proof that it is impossible to quantum sign quantum messages. Although it is indeed impossible to implement (nonarbitrated) quantum signing of quantum messages, it is still possible to accomplish a closely related task: the quantum authentication of quantum messages. In the case of signature protocols the message needs to be readable for everyone; in authentication schemes this condition is relaxed and only a trusted party is able to read the message. In contrast to signing, an authenticated message can be verified through a (secret) key shared between the issuer of the message and its intended recipient. As the authors of Ref. [37] show, authenticated quantum messages must be encrypted, while no such requirement is imposed for classical messages. The above-mentioned research sparked great interest of the scientific community. Quantum signing of classical messages (classical bits) has been studied for instance in Refs. [38 40]. In 013, Li et al. explicitly showed [41] that quantum signature of a quantum message (qubits) is possible in arbitrated schemes. They also showed that this fact does not contradict the impossibility of nonarbitrated quantum signing mentioned before [37]. A number of protocols implementing this task have been derived so far [4 48]. Apart from quantum signing, attention was also focused on closely related protocols, such as quantum authentication of quantum messages [37,49], quantum identification [50], or fingerprinting [51]. The potential for future realistic deployment of quantum signature and authentication can be demonstrated on the basis of two recent experiments [5,53]. Since future quantum networks should be able to benefit from all aspects of quantum information, we will focus on protocols involving a genuine quantum message. A number of possible general approaches has been proposed in this matter. For instance, a complex multiparticle GHZ entangled state distributed to the communicating parties can be used for message signing [48]. Its generation and distribution might, consequently, prove to be experimentally difficult. But other protocols using only Bell states [45] or separable states [44] have been also derived. In these cases however, the sender has to have multiple copies of the quantum message. This might be problematic because in principle quantum information cannot be perfectly copied. Two alternative approaches have been proposed by Barnum et al. [37]. These authors overcome the need for multiple copies of quantum message by inserting additional qubits into the data stream and subsequently perform authenticity verification based on syndrome measurement of a stabilizer purity test code. They have derived two schemes. The first one uses a distribution of entangled Bell states, whereas the second only transmits separable states. There is, however, an inherent problem with the linear-optical implementation of these two schemes. In order to perform a general stabilizer purity test, all transmitted qubits have to be collectively processed by a complex multiqubit quantum circuit. Considering the probabilistic nature of linear-optical quantum gates, this procedure will be highly ineffective and would require additional resources in the form of ancillary photons so that individual quantum gates can be joined together [9]. In this section we propose a feasible quantum authentication protocol for quantum messages that does not rely on entanglement distribution, availability of multiple copies of the message, or complex multiqubit measurements and is, therefore, suitable for the platform of linear optics. At the same time, our scheme is efficient in the sense that only n photons (each encoding two qubits) are necessary to send a n-qubit long quantum message. The general idea behind our scheme is to use two degrees of freedom of light, i.e., one degree of freedom for the message itself and another degree of freedom for a one-time quantum authentication key. If one can decouple the message from the one-time key, one can also attack the scheme by coupling a counterfeited message with a previously decoupled key. There are two possible solutions: (i) the key reflects the message (in classical computing the value of an irreversible function of the message is used as a key), (ii) the message is encrypted in a way that allows only the recipient to verify its authenticity. The first solution, used for instance in [44], requires some information about the message that cannot be provided assuming that no perfect copies of the message exist. Therefore, we resort to the second approach. This approach, as proved in [37], requires sharing a secret key between communicating parties and some sort of encryption of the transmitted message. In our solution, the degree of freedom encoding message or key qubit is chosen at random and this random choice has a similar purpose as message encryption in [37]. Our authentication protocol follows several steps: Assume Alice wishes to transmit a quantum message M composed of n independent qubits m. These qubits are stored in polarization-encoded states of single photons m =α H + β V. Her goal is to transmit this message to Bob using a quantum authentication protocol so that Bob can be sure of its authenticity. Step 1. Bob and Alice establish a standard quantum cryptography communication channel such as the Bennett- Brassard 1984 (BB84) or Renes (R04) [54] protocol. Using this channel, Bob sends two encrypted strings S 1 and S of classical information each n-bit long (referred to as the first and second salt string). This approach is inspired by Assis et al. [55]. Alice generates a random one-time key K composed of n polarization-encoded qubits k. She uses the first salt string S 1 to establish the state preparation basis. If the corresponding bit equals 0, she randomly prepares either diagonal + or antidiagonal polarization states [56]. If the salt bit equals 1, she randomly chooses right ( R ) orleft ( L ) circular polarization states. Step. Alice takes the first qubit of the message ( m ) and the first qubit of the key ( k ). These two qubits are subsequently inserted into a quantum router. The router encodes the control qubit into the spatial mode of a signal photon while preserving the message in its polarization state. For reference see [6]. If the first bit of Bob s second salt string S is 0, then m serves as signal information and k is the control qubit. Then, the router output state becomes 1 [(α H +β V ) 1 ± i r (α H +β V ) ], (9) where the ± sign depends on Alice s choice and the value of r corresponds to the value of the relevant bit in the first salt S-33

108 BARTKIEWICZ, ČERNOCH, AND LEMR PHYSICAL REVIEW A 90, 0335 (014) string. On the other hand, if the first qubit of Bob s second salt is 1, then the roles of m and k are reversed ( k becomes signal and m becomes control). This swap leads to the output state 1 [α( H ±i r V ) 1 + β( H ±i r V ) ]. (10) Alice then sends the output qubit encoded both in polarization and spatial mode to Bob. Step 3. Bob receives the photon from Alice and, depending on the value of his second salt string S bit, he either performs an encoding swap or not. This is a simple encoding swap operation that transforms (10) into(9) and vice versa. Subsequently, Bob subjects the photon to the decoupling procedure described in Sec. II. As demonstrated in that section, Bob obtains the message qubit from the ancillary output port of the decoupling device while he simultaneously performs a projection measurement either on + and or R and L polarizations in key output port depending on the value of the respective bit of the first salt string S 1. Step 4. After Bob s successful detection, he asks Alice to publish her choice of key state k using a quantum authenticated classical channel and compares that to his measurement outcome. All subsequent message qubits are transmitted in the same way. It is imperative that Alice uses her authentication key K only once. Note that the above-mentioned protocol can be implemented even without a quantum router. One can send the key and message qubits separately. Because the second salt string is used, the order in which they are sent is randomized. On the other hand, the quantum router permits the encoding of both qubits into one physical photon, making the transmission more effective. To demonstrate this effect, consider a transmission line with transmissivity τ. A state of two consecutive photons is then transformed according to km τ km, where k and m denote key and message qubits stored in individual photons. Since the protocol succeeds only if both the photons reach Bob s end, we have postselected only this case. It occurs with success probability of τ 4. On the other hand, if one uses the router and thus encodes the information into both polarization and spatial degrees of freedom of one single photon, the state can be expressed in the form of (9). Without loss of generality, let us consider the key qubit being in the state + and being encoded into the spatial mode. The transformation of this state imposed by the lossy channel now reads 1 ( m0 + 0m ) τ 1 ( m0 + 0m ). It follows from this equation that the success probability (the probability of successful transport) is now τ which is a significant improvement over the nonrouter based strategy especially for small channel transmissivities. A. Message falsification attempt Let us begin our analysis by considering that Eve has no aprioriinformation about the message sent by Alice to Bob. If Eve attempts to falsify it (replace it with her own message), she has to guess which of the two possibilities (9) or (10) has been selected (because she does not know the salt string). If she guesses correctly (in half of the cases), she can falsify the message qubit. If she guesses incorrectly, she will mistakenly use the message m as the key. In this case, there is a 1 possibility that Bob will detect an error (assuming the message is random). Therefore, Eve trying to falsify the information introduces an inherent error rate ɛ = 1 4.Forn number of message qubits, the probability of Eve managing to counterfeit n qubits and not be detected is given by ( ) 3 n P C = (1 ɛ) n =. (11) 4 For example, if n equals 10 it is about 6%. For n equal to 0, it would be as low as 0.3%. A problem can arise when the message is too short, or if Eve only tries to counterfeit a small fraction of the message. To resolve this issue Alice and Bob can introduce decoy states. As a decoy state Alice sends an orthogonal state to the current key state instead of the real message qubit. After Bob receives this state, she informs him that it was a decoy and Bob performs verification measurement. In this case even if Eve guesses correctly the salt string, there is a 1 probability that Bob will detect an error by measuring polarization of the message qubit replaced by Eve with falsified data. The overall probability of successful counterfeiting k qubits from a (n + d)-qubit string is now given as ( ) 3 k d ( n ) m k m P C (n,d,k,m) = ( ), (1) 4 k m n+d k where d is the number of randomly inserted decoy states and m is the number of decoy states attacked. All qubits are successfully counterfeited with probability P C (n,d,n + d,d) = ( 3 4 )n ( 3 8 )d. This probability drops rapidly with the length of the transmitted string. It is also useful to investigate the average conditional fidelity of the transmitted chain of qubits, where the condition is successful authentication. Let us therefore consider an n-qubit long message with the addition of d decoy qubits, where k n + d qubits are altered by Eve using the above described strategy. The unaltered qubits are transmitted with fidelity equal to 1 while the counterfeited ones are transmitted with an average fidelity of 1 (random states). As a result the fidelity of the transmission is given by F (k,m,n) = 1 k m (13) n and with probability P C (n,d,k,m) this transmission passes authenticity verification. Eve can attack an arbitrary number of qubits k while successfully attaching m decoy states. Thus, the scheme works on average with the mean fidelity n+d k=0 F n,d = g(k) m f m=m i P C (n,k,m,d)f (k,m,n) n+d k=0 g(k) m f m=m i P C (n,d,k,m) = 1 η, (14) S-34

109 USING QUANTUM ROUTERS TO IMPLEMENT QUANTUM... PHYSICAL REVIEW A 90, 0335 (014) where g(k) is a function describing the probability of Eve attacking k qubits (for the sake of simplicity we assume that g = 1, i.e., uniform distribution), m i = max[0,k n], m f = min[d,k], and η = n+d mf k=0 n+d k=0 m=m i P C (n,d,k,m) k m n mf. (15) m=m i P C (n,d,k,m) Let us also define the average probability of counterfeiting as P n,d = n+d k=0 mf m=m i P C (n,d,k,m) n+d k=0 mf. (16) m=m i An inspection of Eq. (15) reveals that for the limit of long messages (n ), the value of η becomes zero and the average fidelity reaches asymptotically the value of 1 (see Fig. 4). Note that by adding a sufficient number of decoy states to the transmitted qubit stream one can asymptotically achieve η 0. This is because, for a large number of decoy states d n, it is more likely for decoy states to be attacked rather than the states forming the message [see Eq. (1) while ignoring the 3 k 4 k m factor]. In other words, for large values of d it is very likely that k m. This means that the majority of errors will be found in the decoy states. Second, for a large number of decoy states d, the probability of successful counterfeiting [see Eq. (1), while ignoring the probability of Eve attacking m decoy states and k m message qubits] is significant only if k 1. Thus, the mean over k is negligible. The average probability of counterfeiting for a fixed value of n drops rapidly with d (see Fig. 5). So far we have worked under the assumption that Eve is unfamiliar with the content of the transmitted message. Let us now extend our analysis and assume that Eve managed to acquire complete information about the message and wishes to counterfeit it. Note that, since we are not dealing with cryptography but with authentication, the message itself is F n,d d FIG. 4. (Color online) The average transmission fidelity F n,d givenbyeq.(14) for undetected individual counterfeiting of n-qubit long messages extended by d decoy qubits. The fidelity is averaged over all possible cases of successful counterfeiting of extended message which includes attacks on 1 to n + d qubits. n P n,d d FIG. 5. (Color online) Same as in Fig. 4 but for the average counterfeiting probability P n,d given by Eq. (16). not a secret. Eve s strategy now consists of decoupling the message m from the key k qubits. At the time Eve still does not know the salt string, and so is unable to tell which qubit is the message and which is the key. She then picks one of the qubits at random and performs a projective measurement = m m m m, (17) where m is a qubit state orthogonal to m. Whenever this projection measurement yields the value of 1, Eve discards the entire operation and the lost photons would appear to Bob as mere technological losses. There is a 1 probability that Eve correctly picked the message qubit and subjected it to the projection measurement. In this case the measurement outcome is always +1 and Eve uses the remaining key qubit to authenticate a forged message qubit. When this happens, Bob does not register any errors (ɛ = 0). On the other hand, there is a 1 probability that Eve incorrectly picked the key qubit and subjected it to the projection measurement. Since there is no correlation between the key and message qubits, the measurement outcome would yield +1 or 1 with balanced probabilities. Considering that outcomes 1 are discarded, the overall probability of Eve sending an incorrect key is reduced by. Assuming that, similarly to the previous analysis, Bob detects the incorrect key in half of the cases, the error rate introduced by Eve s wrong choice is ɛ = 1 4. In total, the average error rate (considering both good and bad guesses) introduced by Eve using this strategy is ɛ = 1 8 and the probability of successfully counterfeiting n-qubit long message without being detected reads ( ) 7 n P C = (1 ɛ) n =. (18) 8 Again, Alice can insert decoy qubits at random to arbitrarily decrease Eve s chances of undetected counterfeiting to P C = n ( ) 7 n ( ) 1 d, (19) S-35

110 BARTKIEWICZ, ČERNOCH, AND LEMR PHYSICAL REVIEW A 90, 0335 (014) at the expense of lowering the transmission rate. The average fidelity in this scenario can be similarly analyzed. The result would not differ quantitatively from the ones in the preceding paragraph. Also note that Eve can adopt a strategy that is more complex than simple projection (17), but the analysis of this possibility is beyond the scope of this paper. However, for any individual attack strategy all the transmitted qubits can be attacked successfully with probability P C proportional to p d decoy 0ford 1, where p decoy < 1 is the probability of keeping a decoy state unaltered. This means that the malicious behavior is possible only if Eve knows the location of all the decoy states. B. Adding noise to the message The security of this protocol depends on the symmetry between key and message qubits. Eve s inability to decide which qubits belong to the key and which are the message makes it impossible for her to affect one without affecting the other. The message is not secret. However, if Eve tries to read it she will introduce errors in the key as well. A similar situation occurs when Eve decides to undertake a hostile action to discredit Alice by adding noise to her message without actually having any information about its content. In this scenario, Eve malevolently changes the message by applying a random transformation unknown to Alice or Bob while the message itself is not known to Eve. Here we demonstrate that such action can be detected by Bob if he analyzes the key. Each of the qubits is assumed to be attacked independently. Even though we restrict our analysis to individual attacks, it is likely that the results hold even for collective attacks, since the message qubits are not correlated. Eve does not know if the message is represented by a spatial degree of freedom or the polarization state of a photon. In order to alter the message only, Eve has to guess the way in which its qubits are encoded. If she guesses correctly she will not be detected when changing the attacked qubit. She can choose to attack the spatial degree of freedom by tampering with one rail only, or she can attack the polarization degree of freedom by applying the same optical transformation in both rails. Both these strategies are equivalent. This is because Eve cannot always alter a message qubit m without being detected. If she attempts to change the qubit then, with probability 1,she will alter a key qubit k instead. Any single-qubit operation performed by Eve can be generated by Pauli s matrices σ n for n = x,y,z. Thus, if σ n k < 1 for each n, an error can be detected at the last step of the protocol revealing Eve s presence. The key qubit is given as k = 1 ( H ±i r V ). (0) One can show that δ r,0 for n = x σ n k = δ r,1 for n = y 0 for n = z, (1) where δ i,j is Kronecker s δ. Alice alters the value of x at random. This means that if Eve decides to perform σ x or σ y operation, her action will be detected with probability 1 4. However, in the case of σ z she will introduce errors with probability 1. Hence, an eavesdropper cannot manipulate an individual message qubit without introducing errors that can be detected at the last step of the protocol. Note that there is some asymmetry between the xy plane and the z direction. This asymmetry could be removed by using all six eigenstates of Pauli s matrices as possible key qubits. However, as we have shown above, this is not necessary. The above analysis demonstrates the rationale behind using two sets of eigenstates of either σ x or σ y as potential key qubits. If we decided to use only one basis, e.g., r = 0, those operations that can be described as rotations around the x axis would not be detected. This would leave Bob unaware of Eve tampering with the quantum message. Another possible way around it would be to force Alice to send only eigenstates of σ x operator as a message. However, this would make the message classical and the protocol would be secure only with classical information. According to the analysis presented in this subsection, Eve causes authentication errors if she tries to malevolently forge the transmitted message even without actually knowing it or replacing it by her own message. Therefore, average fidelity can be calculated on the same premises, and with the same results as in Eq. (14). IV. ROUTER ASSISTED BELL-STATE DISCRIMINATION AND NONDESTRUCTIVE MANIPULATION Bell-state discrimination is a procedure used to project a two-qubit state onto one of the four maximally entangled Bell states ± 1 ( 00 ± 11 ), (a) ± 1 ( 01 ± 10 ). (b) This procedure is crucial to several quantum information algorithms, including quantum teleportation [57], entanglement swapping [58], or dense coding [59]. On the platform of linear optics, this task is problematic because of the probabilistic nature of single-photon behavior on beam splitters. A balanced beam splitter and polarization projection can be used to discriminate between polarization encoded singlet state and one of the triplet states + [60,61]. It would however be impossible to distinguish the remaining two triplet states + and even if photon number resolving detectors are used. In general, deterministic and complete Bell-state discrimination is impossible with linear optics [6]. Therefore, one has to resort to either unambiguous protocols or go outside the scope of linear optics [63]. A number of approaches to full Bell-state discrimination have been published. For instance, additional photon ancillae can be used to perform this task within linear optics [64]. Alternatively, one can consider more degrees of freedom than polarization by using hyperentanglement [65]. An experimental demonstration of this procedure using polarization and orbital angular momentum has been presented in 007 [66]. A linear-optical controlled-phase gate can serve as an probabilistic disentangling gate that transforms Bell states into mutually orthogonal and therefore distinguishable separable S-36

111 USING QUANTUM ROUTERS TO IMPLEMENT QUANTUM... PHYSICAL REVIEW A 90, 0335 (014) two-photon states [67]. It is also possible to implement full Bell-state discriminator using experimentally challenging nonlinear optical phenomena [68]. Also, several schemes for the discrimination of higher dimensional entangled states have been implemented [69]. In this section we describe how a quantum router can assist in performing a complete Bell-state discrimination. Furthermore, we extend our analysis to include complete nondestructive Bell-state manipulation, allowing for coherent engineering of a two-qubit state directly in the Bell basis. In principle this task cannot be achieved by any of the above-mentioned techniques. So far we have only considered mutually separable signal and control states. This assumption is valid if the router is a tool for steering the signal. Nevertheless, the router can deal with general signal and control input quantum state in the form of ψ s ψ c IN,C = c c 01 +c c (3) Recalling the transformation implemented by the router (see Sec. II), such input state gets transformed into ψ s ψ c IN,C c c 0 + c c 4 1, (4) where lower indices denote the spatial mode of the output signal. Let us now focus solely on the platform of polarization qubits and linear optics. The input state (3) can be encoded using the horizontal and vertical polarization of individual photons, ψ s ψ c IN,C = c 1 HH +c HV +c 3 VH +c 4 VV. (5) Alternatively, one can express the input state in terms of Bell states (Bell-states basis) ψ s ψ c IN,C = α 1 +α + +α 3 +α 4 +. (6) It can be seen that the router processes all Bell states individually, transcribing them into single-photon output state encoded in both polarization and spatial mode ± IN,C 1 ( H 1 ± V ), (7a) ± IN,C 1 ( H ± V 1 ), (7b) where the lower indices denote spatial modes of the output. To distinguish between these four output states an additional setup (depicted in Fig. 6) is added to the output of the router. It consists of a Bell-state splitter and a set of four single-photon detectors. In the first step, we separate the states ± from ± by combining the two output modes on a polarizing beam splitter (transmitting horizontal and reflecting vertical polarization). If the signal and control qubits of the router were in one of the ± ( ± ) states, the photon from output of the router travels to the upper (lower) mode of the first PDBS shown in Fig. 6. Hence, the combined transformation of the router followed by Bell splitting reads ± IN,C 1 ( H ± V ) 1 = ± 1, (8a) ± IN,C 1 ( H ± V ) = ±. (8b) 1 FIG. 6. (Color online) Router assisted Bell-state discriminator as described in the text: PBS1, polarizing beam splitter transmitting horizontal and reflecting vertical polarization; PBS and PBS3, beam splitters transmitting diagonal and reflecting antidiagonal linear polarizations; DET, detection block with four detector each projecting onto one of the four Bell states. In order to further distinguish between different Bell states we split each of the output modes in a diagonal or antidiagonal linear polarization basis. This can be performed by using a rotated polarizing beam splitter (transmitting diagonal and reflecting antidiagonal linear polarization) or by implementing a Hadamard transform by a half-wave plate followed by a PBS splitting H and V polarization components. As depicted in Fig. 6, full Bell-state discrimination is performed by detecting a photon by one of the four detectors which corresponds to a projection in Bell s basis. A. Arbitrary Bell basis state manipulation In this subsection we explain how a quantum router can be used to perform arbitrary operations in Bell s basis. In contrast to Bell-state discrimination, quantum state manipulation does not project the quantum state on one of the Bell states and, thus, maintains coherence. This task cannot be performed (at least not without restrictions) with just one controlled-phase gate. The controlled-phase gate transforms Bell states into separable mutually orthogonal two-photon states. These states remain correlated. It is, for instance, impossible to filter out just one of them from a two-photon state by applying local polarizationsensitive attenuation because a pair of states would be affected. We have already described the principles behind routerassisted decomposition of a quantum state into Bell states. As illustrated in Fig. 7, the set of four detectors can be replaced by a set of four neutral density filters and phase shifters. The parameters of these filters and phase shifters can be set independently. Hence, after applying inverse of the Bell splitter depicted in Fig. 6 followed by the quantum decoupler we can implement an arbitrary transformation IN,C τ 1 e iϕ1 S,A, (9a) + IN,C τ e iϕ + S,A, (9b) IN,C τ 3 e iϕ3 S,A, (9c) + IN,C τ 4 e iϕ4 + S,A, (9d) where τ m and ϕ m for m = 1,,3,4 are independent amplitude transmittances and phase shifts, respectively. Note that this procedure allows any two-qubit state in Bell basis to be directly engineered without disturbing its coherence. Using the S-37

112 BARTKIEWICZ, ČERNOCH, AND LEMR PHYSICAL REVIEW A 90, 0335 (014) FIG. 7. (Color online) Bell-state manipulation using quantum router. The two-qubit state is processed by a quantum router. One of the qubits acts as signal, the other as control. Subsequently, the output signal is split on Bell-state splitter depicted in Fig. 6. Set of four neutral density filters and phase shifters is used to adjust amplitude and phase of each individual Bell state separately. The signal is recombined on an inverse Bell-state splitter and finally two separate photons are recreated using the quantum decoupler described in Fig.. above-described components one can implement an arbitrary positive valued measurement in the Bell s basis. This is especially interesting for investigating nonlinear properties of quantum states [70 7]. V. CONCLUSIONS In this paper we have analyzed the properties of quantum routers and their corresponding inverse devices quantum decouplers. We have demonstrated how quantum routers can be used to implement prominent quantum information protocols on two examples: a quantum authentication procedure for the authentication of quantum messages, and the use of a quantum router for nondestructive complete Bellstate manipulation. Both of these examples demonstrate that quantum routers should not be reduced to quantum analogues of classical routers, but rather considered to be important and versatile components of future quantum networks. The recent experimental demonstration of a linear-optical quantum router [7] makes the theoretical concept of quantum routing experimentally feasible. This paper proposes viable options, which should be considered when dealing with practical quantum authentication or nondestructive Bell-state manipulation. ACKNOWLEDGMENTS K.B. gratefully acknowledges the support by the Foundation for Polish Science, the Operational Program Research and Development for Innovations European Regional Development Fund (Project No. CZ.1.05/.1.00/ ) and the Operational Program Education for Competitiveness European Social Fund (Project No. CZ.1.07/.3.00/0.0017) of the Ministry of Education, Youth and Sports of the Czech Republic. K.L. acknowledges the support by Czech Science Foundation (Grant No P). A.Č. acknowledges the Czech Science Foundation (Grant No. P05/1/038). This work was also supported by the Polish National Science Centre under Grant No. DEC-011/03/B/ST/ [1]M.A.NielsenandI.L.Chuang,Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 000). [] D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, Heidelberg, 001). [3] D. Bruß and G. Leuchs, Lectures on Quantum Information (Wiley-VCH, Berlin, 006). [4] N. Gisin, S. Pironio, and N. Sangouard, Phys.Rev.Lett.105, (010). [5] D. Pitkanen, X. Ma, R. Wickert, P. van Loock, and N. Lütkenhaus, Phys. Rev. A 84, 035 (011). [6] M. Curty and T. Moroder, Phys.Rev.A84, (R) (011). [7] M. Mičuda, I. Straka, M. Miková, M. Dušek, N. J. Cerf, J. Fiurášek, and M. Ježek, Phys. Rev. Lett. 109, (01). [8] S. Kocsis, G. Y. Xiang, T. C. Ralph, and G. J. Pryde, Nat. Phys. 9, 3 (013). [9] M. Bula, K. Bartkiewicz, A. Černoch, K. Lemr, Phys. Rev. A 87, (013). [10] E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein, and K. Lemr, Phys.Rev.A88, 0137 (013). [11] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (00). [1] K. Bartkiewicz, K. Lemr, A. Černoch, J. Soubusta, and A. Miranowicz, Phys.Rev.Lett.110, (013). [13] R. Ursin et al., Nat. Phys. 3, 481 (007). [14] S. Wang, W. Chen, J.-F. Guo, Z.-Q. Yin, H.-W. Li, Z. Zhou, G.-C. Guo, and Z.-F. Han, Opt. Lett. 37, 1008 (01). [15] L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, Nat. Photon. 4, 686 (010). [16] H.-K. Lo, M. Curty, and B. Qi, Phys. Rev. Lett. 108, (01). [17] P. W. Shor, Phys.Rev.A5, R493 (1995). [18] A. R. Calderbank and P. W. Shor, Phys.Rev.A54, 1098 (1996). [19] A. M. Steane, Phys.Rev.Lett.77, 793 (1996). [0] D. Gottesman, Phys.Rev.A57, 17 (1998). [1] J. Modlawska and A. Grudka, Phys. Rev. Lett. 100, (008). [] A. Paetznick and B. W. Reichardt, Phys. Rev. Lett. 111, (013). [3] Y. S. Weinstein, Phys.Rev.A88, 0135 (013). [4] M. A. Hall, J. B. Altepeter, and P. Kumar, Phys.Rev.Lett.106, (011). [5] K. Lemr and A. Černoch, Opt. Commun. 300, 8 (013). [6] K. Lemr, K. Bartkiewicz, A. Černoch, and J. Soubusta, Phys. Rev. A 87, (013). [7] C. Vitelli, N. Spagnolo, L. Aparo, F. Sciarrino, E. Santamato, and L. Marrucci, Nat. Photon. 7, 51 (013). [8] D. Zueco, F. Galve, S. Kohler, and P. Hänggi, Phys. Rev. A 80, (009). [9] T. Aoki, A. S. Parkins, D. J. Alton, C. A. Regal, B. Dayan, E. Ostby, K. J. Vahala, and H. J. Kimble, Phys. Rev. Lett. 10, (009) S-38

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115 PRL 114, (015) P H Y S I C A L R E V I E W L E T T E R S week ending 17 APRIL 015 Experimental Implementation of Optimal Linear-Optical Controlled-Unitary Gates Karel Lemr, 1,* Karol Bartkiewicz,,1, Antonín Černoch, 3, Miloslav Dušek, 4 and Jan Soubusta 3 1 RCPTM, Joint Laboratory of Optics of Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic, 17. listopadu 1, Olomouc, Czech Republic Faculty of Physics, Adam Mickiewicz University, PL Poznań, Poland 3 Institute of Physics of Academy of Sciences of the Czech Republic, Joint Laboratory of Optics of PU and IP AS CR, 17. listopadu 50A, Olomouc, Czech Republic 4 Department of Optics, Faculty of Science, Palacký University, 17. listopadu 1, cz Olomouc, Czech Republic (Received 16 October 014; published 16 April 015) We show that it is possible to reduce the number of two-qubit gates needed for the construction of an arbitrary controlled-unitary transformation by up to times using a tunable controlled-phase gate. On the platform of linear optics, where two-qubit gates can only be achieved probabilistically, our method significantly reduces the amount of components and increases success probability of a two-qubit gate. The experimental implementation of our technique presented in this Letter for a controlled single-qubit unitary gate demonstrates that only one tunable controlled-phase gate is needed instead of two standard controlled- NOT gates. Thus, not only do we increase the success probability by about 1 order of magnitude (with the same resources), but also avoid the need for conducting quantum nondemolition measurement otherwise required to join two probabilistic gates. Subsequently, we generalize our method to a higher order, showing that n-times controlled gates can be optimized by replacing blocks of controlled-not gates with tunable controlled-phase gates. DOI: /PhysRevLett PACS numbers: 4.50.Ex, Lx, 4.50.Dv Quantum computing is a promising direction in information processing [1,]. Similarly to classical computing, quantum circuits are composed of various elementary gates. In 1989, Deutsch proved the existence of a universal threequbit gate [3]. Later, DiVincenzo showed that Deutsch s gate can be implemented by a sequence of two- and singlequbit gates [4]. Meanwhile, Barenco discovered a class of two-qubit gates sufficient for building any quantum circuit [5]. A practical set of universal gates was defined later [6]. This set of gates includes several single-qubit gates and only one two-qubit gate the controlled-not (CNOT) gate. Although the method presented by Barenco et al. [6] shows how to construct any quantum circuit, it does not take into account various optimization procedures [7 17]. Optimization is crucial in linear optics, where the CNOT gate can only be implemented probabilistically [18 1], meaning that every repetition reduces the success probability of the entire scheme. In 009, Lanyon et al. demonstrated a considerable reduction in the number of CNOT gates necessary for circuit construction by introducing additional ancillary modes []. They have constructed a Toffoli gate (controlledcontrolled-cnot gate) with only two CNOT gates and have also designed a generalized controlled-phase gate, but not an optimal one. Mičuda et al. presented a method further reducing resources needed for the implementation of a Toffoli gate to only one CNOT gate [3]. This reduction is achieved by combining polarization and spatial encoding to encode a two-qubit state into one single photon. However, the preparation of a specialized control two-qubit state is problematic. It is possible to use quantum routers (or quantum state fusion) [4 6], but this would mean using additional CNOT gates, which would cancel the achieved reduction. Until recently only approaches involving standard CNOT gates [7 16] or their nonoptimal generalizations [17,] were considered. In 010, Kieling et al. proposed an optimal (without auxiliary photons) linear-optical implementation of a tunable CPHASE gate that imposes a given tunable phase shift φ [7] jkli e iφδ k1δ l1jkli; ð1þ where k and l take values of logical qubit states 0 or 1 and δ is the Kronecker s delta. In 011, the first optimal tunable CPHASE gate was experimentally demonstrated [8]. The experiment also allowed us to verify and explain the optimal success probability of the gate as a function of the phase shift φ [7,9]. In this Letter we show that using a tunable CPHASE gate instead of a CNOT gate makes it possible to (i) reduce the complexity of various quantum circuits and (ii) increase the success probability of these circuits in linear optics. The support for our idea comes from an experimental implementation of the proposed scheme. Arbitrary single-qubit controlled-unitary transformation. It has been shown by Barenco et al. [6] that two controlled-sign gates are needed to implement an arbitrary controlled-unitary operation acting on a signal qubit and controlled by a control qubit. In special cases, one =15=114(15)=15360(5) American Physical Society S-41

116 PRL 114, (015) P H Y S I C A L R E V I E W L E T T E R S week ending 17 APRIL 015 controlled-sign gate is sufficient, but at the expense of restricting the class of implemented operations. Considering the probabilistic nature of controlled-sign gates on the platform of linear optics, it is crucial to limit their repetition as much as possible. We show that only one single tunable controlled-phase gate is needed for the construction of a universal single-qubit controlled-unitary operation. Note that the success probability of two consecutive controlled-sign gates would be 1=81 (using linear optics only and no photon ancillae), the minimum success probability of a tunable controlled-phase gate is 1=11 (0.14 on average). Moreover, by reducing the number of gates from two to one, we also avoid the need for intermediary nondemolition presence detection otherwise required to join two probabilistic gates [30,31]. Let us consider the scheme depicted in Fig. 1. While the upper (control) qubit undergoes only the controlled-phase operation, the lower (signal) qubit is subjected to a set of unconditional single-qubit gates before and after it enters the controlled-phase gate. These unconditional singlequbit gates can be implemented deterministically on the platform of linear optics. The initial set of single-qubit operations consists of one rotation in the z direction e iα= 0 ZðαÞ ¼ 0 e iα= followed by another rotation in the y direction ðþ cos θ sin θ YðθÞ ¼ : ð3þ sin θ cos θ Similarly, the single-qubit rotations inserted behind the controlled-phase gate are Yð θþ and Zð αþ. When the control qubit is j0i, the controlled-phase gate does not impose any phase shifts and all unconditional single-qubit rotations cancel each other Zð αþyð θþyðθþzðαþ ¼1: On the other hand, if the control qubit is j1i, the controlled-phase gate introduces an additional rotation FIG. 1 (color online). Quantum computation circuit implementing an arbitrary single-qubit controlled-unitary operation W [see Eq. (4)] by means of one tunable controlled-phase gate and several unconditional single-qubit operations. in the z direction ZðφÞ [see Eq. ()]. The overall operation imposed on the signal qubit now reads W ¼ Zð αþyð θþzðφþyðθþzðαþ: ð4þ To demonstrate the universality of the above mentioned gate, let us consider the following: Any single-qubit unitary transformation can be described as a rotation along some axis on the Bloch sphere that corresponds to an operator R ψ ðφþ ¼e iφ= jψihψjþe iφ= jψ ihψ j; ð5þ where φ denotes the rotation angle and jψi is the state that geometrically corresponds to the rotation axis on the Bloch sphere (jψ i is orthogonal to jψi). For rotation along the z direction we have jψi ¼j0i, which inserted into Eq. (4) yields W ¼ Zð αþyð θþðe iφ= j0ih0jþe iφ= j1ih1jþyðθþzðαþ: ð6þ Using prescriptions () and (3) we can easily verify that jψi¼zð αþyð θþj0i¼e iðα=þ cos θ j0iþe iðα=þ sin θ j1i; ð7þ and thus show that any arbitrary pure qubit state is accessible if suitable values of α and θ are set. Unitary transformations maintain orthogonality so that j1i Zð αþyð θþj1i ¼jψ i. The two pairs of unconditional single-qubit rotations before and after the CPHASE gate permit for any rotation axis. Tuning the phase of the CPHASE gate permits for setting any rotation angle φ. Any single-qubit unitary operation can be decomposed in the form of [6] U ¼ ZðγÞYðωÞZðδÞ; ð8þ parametrized by three real numbers. An explicit decomposition of the transformation matrix can be found in the Supplemental Material [3]. Note that the matrix W in Eq. (4) is also parametrized by three real numbers. Optimality of our method is guaranteed by the fact that we only use one probabilistic gate that is optimal for any given phase shift required by the transformation. Experimental implementation. We have constructed an experimental setup as depicted in Fig.. It consists of a tunable CPHASE gate placed between single-qubit gates in the signal mode that implement the required unconditional rotations Z and Y. In our experiment we encode qubits into polarization states of individual photons (j0i corresponds to horizontal polarization jhi, j1i to vertical polarization jvi). Unconditional single-qubit rotations Z and Y are implemented by sets of one half-wave and one quarter-wave S-4

117 PRL 114, (015) P H Y S I C A L R E V I E W L E T T E R S week ending 17 APRIL 015 FIG. (color online). Schematic drawing of the experimental setup. The components are labeled as follows: MT motorized translation, HWP half-wave plate, QWP quarter-wave plate, PBS polarizing beam splitter, BDA beam divider assembly, BD beam divider, F neutral density filter, D detector. For more details on the setup function, see the Supplemental Material [3]. plates. The control state preparation is achieved by one half-wave plate in control mode since only logical states j0i and j1i are required. Photons were generated using type I spontaneous parametric down-conversion in a LiIO 3 crystal pumped by 00 mw cw Kr þ laser beam. The observed coincidence rate was ranging approximately from 300 to 3000 counts per sec depending on the success probability given for various settings of φ. By following the procedure described in Ref. [8], we have adjusted the tunable CPHASE gate to a given phase shift φ. We have tested our device on six combinations of tunable CPHASE gate phase shifts φ and single qubit rotations ZðαÞ and YðθÞ (see Table I). In all these six cases, we have performed complete process tomography of the signal mode for the control qubit set to state j0i and then also to j1i [33 36] (for more details on this procedure, please consult the Supplemental Material [3]). The estimated Choi matrices were compared to theoretical predictions permitting to calculate their fidelities F and purities P. We adopt the following labeling: F off and P off stand for fidelity and purity observed with the control qubit set to j0i, while F on and P on denote the same parameters for control qubits in the state j1i. We have also determined the resulting success probabilities by comparing the coincidence rate observed after adjusting the gates with the coincidence rate behind the same setup, but with all filters removed and polarizations set so that no single or twophoton interference takes place. Thus, we obtain the experimental success probability p succ corrected for technological losses (e.g., components back-reflections or coupling losses). The results of our experiment are summarized in Table I and one selected case is also depicted in Fig. 3. Estimated fidelities and purities are typically about 90%. The fluctuations in obtained fidelities and purities are caused by experimental imperfections (e.g., optical components imperfections, adjustment errors) that have different impact for different settings of the setup. In all the cases, the obtained fidelities and purities are sufficiently high to prove the successful implementation of our proposal. Additionally, we have also tested the quantum nature of our controlled-phase gate by measuring the output state negativity [3,37,38] when inserting a separable input state 1 ðj0iþj1iþðj0iþj1iþ. For example, we obtained negativity (theoretical value is 0.191) for φ ¼ π=. The full report on this verification test can be found in the Supplemental Material [3]. n-times controlled single qubit unitary transformations. As in the case of a CNOT gate, a Toffoli gate (CNOT) can be used to implement a controlled unitary gate, but with two control qubits. The CCNOT operation can be implemented using only CNOT gates. We can also build a -times controlled gate by replacing the CNOT gates acting on the target qubit with CPHASE gates and single-qubit rotations. This approach ensures efficiency higher than in the case of a circuit using only CNOTs, where CNOT gates modifying the target qubit are each replaced with a singlequbit rotation sandwiched between two CNOT gates. This means the latter approach is equivalent to adding 3 two-qubit gates to the circuit proposed in Lemma 6.1 in Ref. [6]. The situation is analogous for any n-times controlled unitary gate. Designing an arbitrary n-times controlled TABLE I. Experimental results for various settings of the CPHASE gate parameter φ together with single qubit rotations α and θ. The corresponding values of standard decomposition parameters ω, γ, and δ are also calculated. F off and P off denote estimated process fidelity and purity of the transformation with the control qubit set to j0i, while F on and P on denote the same characteristics with the control qubit set to j1i. p succ and p succth stand for experimental and theoretical success probability. φ θ α ω γ δ F off P off F on P on p succ p succth π=8 π= 0 π=8 π= π= π=4 0 π= 0 π=8 π= π= π= π= π= π=4 π= 0 3π=4 π= π= π 0 π= 0 π= π= S-43

118 PRL 114, (015) P H Y S I C A L R E V I E W L E T T E R S week ending 17 APRIL 015 FIG. 3 (color online). Estimated process matrices for φ ¼ð3π=4Þ, θ ¼ðπ=Þ and α ¼ 0 (a) with control qubit j0i and (b) with control qubit j1i. Moduli of matrix elements are visualized by bar heights and their phase by arrow directions. The experimentally acquired density matrices are compared to their theoretical counterparts in the Supplemental Material [3]. gate is usually considered in the context of Toffoli gates. However, constructing such circuits directly with Toffoli gates has been proven inefficient as it needs an order of n two-qubit gates [1]. It has been demonstrated that, by extending the Hilbert space of the target information carrier (see Ref. []), one can implement an arbitrary n-times controlled gate by using n 1 standard two-qubit gates performing controlled-pauli operations (R n ðπþ for n ¼ x; y; z). This is considered to be the most effective currently known solution. In linear optics we can increase the efficiency of an n-times controlled unitary gate proposed by Lanyon et al. [] by replacing two standard controlled-pauli gates with a controlled-unitary operation performed with a higher probability of success. In the original scheme, the authors of Ref. [] show that any n-qubit controlled gate can be implemented using only n CNOT gates and single-qubit operations. Similarly, we show in Fig. 4 any n-times controlled-u gate can be reduced to ðn 1Þ CNOT gates and one single-qubit controlled-phase gate. This is possible by using the ðn þ 1Þ-level system as the bottom-most circuit line. This reduces the number of the required twoqubit gates by one, and has the additional merit in having the single qubit controlled-u gate work with a higher average success probability than a CNOT gate. This optimization can be applied in a linear-optical implementation of the -times controlled unitary gate from Ref. []. The improvement is apparent if we consider replacing the product of CNOT, CZ, and R gates by a product of two controlled-unitary gates and single-qubit operations. The product of two controlled-unitary gates works on average with a higher success rate than a product of CZ FIG. 4 (color online). -times and 3-times CPHASE [ZðφÞ] gates decomposed with a method of Lanyon et al. [] based on extending the Hilbert space of the target qubit (the two top rows show the idea from Ref. [] to demonstrate the benefits of our approach). The control lines are denoted as C n for n ¼ 1; ; 3 and T denotes the target qubit line. For the (3)-times controlled gate the dimension of the target qubit is extended by 1(). The additional dimensions are visualized as additional dashed channels. For technical details on the operation of the X a and X b gates see Ref. []. By iterating the depicted procedure for an n- times controlled gate we conclude that ðn 1Þ CNOTs and a single CPHASE are needed for its implementation. An arbitrary controlled-u gate, where U ¼ A ZðφÞA can be easily implemented using the same circuits by applying A and A single-qubit operations. and CNOT gates. Hence, by using an optimal implementation of a CPHASE gate, we can increase the success rate of the circuit by approximately 1 order of magnitude. The currently known most efficient implementation of the Toffoli gate in terms of the number of two-qubit gates was presented by Mičuda et al. in Ref. [3]. This approach requires only one CZ gate. However, it is possible to obtain an even more efficient circuit for the controlled-u if we replace the CZ with a CPHASE gate instead of the product of two CNOTs and single-qubit rotations. Any n-times controlled operation can be constructed from Toffoli gates and a single-qubit controlled unitary gate regardless of the auxiliary resources. Therefore, we infer that our reasoning and the resulting improvement is valid for all implementations of the CCNOT. Since our idea is not limited only to the two-qubit case it can be used to increase the success probability and scaling of more complex quantum circuits. Conclusions. We have presented a method for the optimization of quantum circuits based on a tunable CPHASE gate. This method decreases the number of controlled operations needed in circuit design and significantly increases the success probability on physical platforms where controlled gates can only be implemented probabilistically (e.g., by about 1 order of magnitude using linear optics). When compared to the method presented in S-44

119 PRL 114, (015) P H Y S I C A L R E V I E W L E T T E R S week ending 17 APRIL 015 Ref. [], our method further decreases the number of CNOT gates by 1 in every controlled-unitary block regardless of the number of control qubits. Since these blocks may be used repeatedly in complex quantum circuits the benefits of our method scale with the size of that circuit. This advantage is especially pronounced on the platform of linear optics, where it has the potential to increase the success probability about 10 times for every controlledunitary block used in a quantum circuit. Also, in comparison to the method in Ref. [], our technique does require smaller extension of Hilbert space. The reduction in the number of CNOT gates and in the size of the Hilbert space makes circuits less complex and thus more experimentally accessible. We have demonstrated the experimental feasibility of our approach on the basis of one experimental case, namely, the single-qubit controlled-unitary operation. The authors thank Jára Cimrman for his helpful suggestions. K. L. acknowledges support by Czech Science Foundation (Grant No P). A. Č. acknowledges support by Czech Science Foundation (Grant No. P05/1/ 038). K. B. acknowledges support by the Foundation for Polish Science and the Polish National Science Centre under Grant No. DEC-013/11/D/ST/0638. The above mentioned authors also acknowledge the Project No. LO1305 of the Ministry of Education, Youth and Sports of the Czech Republic. M. D. acknowledges support from the Palacký University (No. IGA-PrF ). * k.lemr@upol.cz bartkiewicz@jointlab.upol.cz acernoch@fzu.cz [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 000). [] D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, Heidelberg, 001). [3] D. Deutsch, Proc. R. Soc. A 45, 73 (1989). [4] D. P. DiVincenzo, Phys. Rev. A 51, 1015 (1995). [5] A. Barenco, Proc. R. Soc. A 449, 679 (1995). [6] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Phys. Rev. A 5, 3457 (1995). [7] S. S. Bullock and I. L. Markov, Proceedings of the 40th Conference on Design Automation (ACM Press, New York, 003), p. 34. [8] J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, Phys. Rev. Lett. 91, (003). [9] V. V. Shende, I. L. Markov, and S. S. Bullock, Phys. Rev. A 69, 0631 (004). [10] J. J. Vartiainen, M. Möttönen, and M. M. Salomaa, Phys. Rev. Lett. 9, (004). [11] M. Möttönen, J. J. Vartiainen, V. Bergholm, and M. M. Salomaa, Phys. Rev. Lett. 93, (004). [1] F. Vatan and C. Williams, Phys. Rev. A 69, (004). [13] G. Vidal and C. M. Dawson, Phys. Rev. A 69, (R) (004). [14] V. Bergholm, J. J. Vartiainen, M. Möttönen, and M. M. Salomaa, Phys. Rev. A 71, (005). [15] V. V. Shende, S. S. Bullock, and I. L. Markov, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 5, 1000 (006). [16] M. Plesch and Č. Brukner, Phys. Rev. A 83, 0330 (011). [17] J. Fiurášek, Phys. Rev. A 73, (006). [18] H. F. Hofmann and S. Takeuchi, Phys. Rev. A 66, (00). [19] N. K. Langford, T. J. Weinhold, R. Prevedel, K. J. Resch, A. Gilchrist, J. L. O Brien, G. J. Pryde, and A. G. White, Phys. Rev. Lett. 95, (005). [0] N. Kiesel, C. Schmid, U. Weber, R. Ursin, and H. Weinfurter, Phys. Rev. Lett. 95, (005). [1] M. Bartkowiak and A. Miranowicz, J. Opt. Soc. Am. B 7, 369 (010). [] B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O Brien, A. Gilchrist, and A. G. White, Nat. Phys. 5, 134 (009). [3] M. Mičuda, M. Sedlák, I. Straka, M. Miková, M. Dušek, M. Ježek, and J. Fiurášek, Phys. Rev. Lett. 111, (013). [4] K. Lemr and A. Černoch, Opt. Commun. 300, 8 (013). [5] K. Lemr, K. Bartkiewicz, A. Černoch, and J. Soubusta, Phys. Rev. A 87, (013). [6] C. Vitelli, N. Spagnolo, L. Aparo, F. Sciarrino, E. Santamato, and L. Marrucci, Nat. Photonics 7, 51 (013). [7] K. Kieling, J. L. O Brien, and J. Eisert, New J. Phys. 1, (010). [8] K. Lemr, A. Černoch, J. Soubusta, K. Kieling, J. Eisert, and M. Dušek, Phys. Rev. Lett. 106, (011). [9] K. Lemr and A. Černoch, Phys. Rev. A 86, (01). [30] P. Kok, H. Lee, and J. P. Dowling, Phys. Rev. A 66, (00). [31] M. Bula, K. Bartkiewicz, A. Černoch, and K. Lemr, Phys. Rev. A 87, (013). [3] See Supplemental Material at supplemental/ /physrevlett for detailed description of the experimental procedure and auxiliary calculations and measurements, which includes Refs. [8,9,33 38]. [33] A. Jamiołkowski, Rep. Math. Phys. 3, 75 (197). [34] M.-D. Choi, Linear Algebra Appl. 10, 85 (1975). [35] M. Ježek, J. Fiurášek, and Z. Hradil, Phys. Rev. A 68, (003). [36] Quantum State Estimation, Lect. Notes Phys., edited by M. G. A. Paris and J. Řeháček (Springer, Berlin, Heildeberg, 004), Vol [37] K. Życzkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Phys. Rev. A 58, 883 (1998). [38] G. Vidal and R. F. Werner, Phys. Rev. A 65, (00) S-45

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121 Journal of Optics J. Opt. 17 (015) 150 (6pp) doi: / /17/1/150 Scheme for a linear-optical controlled-phase gate with programmable phase shift Karel Lemr 1, Karol Bartkiewicz 1, and Antonín Černoch 1 1 RCPTM, Joint Laboratory of Optics of Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic, 17. listopadu 1, Olomouc, Czech Republic Faculty of Physics, Adam Mickiewicz University, PL Poznań, Poland k.lemr@upol.cz Received 19 June 015, revised 8 September 015 Accepted for publication 10 September 015 Published 15 October 015 Abstract We present a linear-optical scheme for a controlled-phase gate with tunable phase shift programmed by a qubit state. In contrast to all previous tunable controlled-phase gates, the phase shift is not hard-coded into the optical setup, but can be tuned to any value from 0 to π by the state of a so-called program qubit. Our setup is feasible with current level of technology using only linear-optical components. We provide an experimental feasibility study to assess the gate s implementability. We also discuss options for increasing the success probability up to 1/1; this approaches the success probability of an optimal non-programmable tunable controlledphase gate. Keywords: programmable gate, quantum gate, controlled-phase gate, linear optics 1. Introduction Quantum computing is a promising approach, which has the potential to considerably improve computing efficiency [1, ]. It has been demonstrated that any quantum circuit can be decomposed into a set of standard single- and two-qubit gates [3]. While the single-qubit gates represent single qubit rotations, the two-qubit gates make the qubits interact and thus process information. A prominent example of such a twoqubit gate is the controlled-phase gate (or its close relative, the controlled-not gate) [4]. The controlled-phase gate performs the following transformation on the target and control qubit states: 00ñ 00 ñ, 01ñ 01 ñ, 10ñ 10 ñ, 11ñ e i j 11 ñ, ( 1) where 0 and 1 in the brackets stand for logical states of both the target and the control qubits, respectively. The parameter j then denotes the introduced phase shift. There have been a number of experimental implementations of the controlledphase gate achieved on various physical platforms, including nuclear magnetic resonance [5], trapped ions [6], and superconducting qubits [7]. On the platform of linear optics, this gate has been implemented using various schemes [8 10] (for review papers see also [11, 1]). All these implementations, however, only considered phase shift j = p, also known as the controlled-sign transformation. Operating the controlled-phase gate at phase shifts other than π was investigated for the first time in a seminal paper by Lanyon et al (009) [13]. In order to achieve diverse phase shifts, the authors increased the Hilbert space by introducing auxiliary modes. Their implementation, however, does not have optimal probability of success. In 010, Konrad Kieling et al proposed a scheme for an optimal linear-optical C-phase gate with tunable phase shift [14]. In 011, this scheme was experimentally implemented and tested in our laboratory [15]. Both the Lanyon et al [13] scheme and our own [15] have the phase-shift hard-coded by the specific settings of various optical elements. But these limit the adaptability and usefulness of these gates in multipurpose quantum circuits. In order to make quantum circuits more versatile, researchers have proposed so-called programmable gates [16]. Instead of hard-coding the transformations into the experimental setup, these gates have their properties programmed by the quantum state of the so-called program qubit. While it would be necessary to use an infinite amount of classical information, one qubit of quantum information suffices to precisely set a real-valued parameter of a quantum gate (or transformation). Such qubit can also be transmitted /15/150+06$ IOP Publishing Ltd Printed in the UK S-47

122 J. Opt. 17 (015) 150 K Lemr et al Figure 1. Conceptual scheme of a programmable C-phase gate. T, C and P denote target, control, and program ports, respectively. The phase shift j, encoded into the state of the program qubit, is equivalent to the phase shift introduced by the gate (equation (1)). over a quantum channel, thus allowing for remote programming of a quantum gate similar to classical software distribution over computer networks. The quantum transformation in question is basically teleported to its user. As a proof-of-principle, Mičuda et al have constructed a programmable phase gate [17]. This gate introduces a programmable phase shift between the logical states 0ñ and 1ñ of a signal qubit. Thus it achieves programmable single-qubit rotation along one axis. The success probability of this scheme has been recently improved to the theoretical limit of 1/ using feed-forward [18]. In this paper, we propose a linear-optical scheme for a tunable C-phase gate with programmable phase shift. This is not to be confused with the programmable phase gate [17, 18], where only a single-qubit rotation was programmed, while in our case we program a two-qubit operation by means of a third qubit. A tunable C-phase gate is a key ingredient for a number of protocols, including quantum routers [19, 0], quantum state engineering [1, ], and controlled-unitary gates [13, 3]. By adding the programmability, we further develop this important gate, making it a more versatile, and, therefore, an even more powerful tool for quantum information processing. Also note that since single qubit rotations and controlled-phase gate are sufficient to construct any quantum circuit, the proposed programmable controlled-phase gate with already known programmable single-qubit rotations allow, in principle, the construction of any programmable quantum circuit. The conceptual scheme of the proposed gate is shown in figure 1. We adopt the following notation throughout the paper: ytñ, ycñ, and ypñ denote the target, control, and program qubits, respectively. The program qubit takes the form of 1 y P = 0 ei ( ñ - j 1 ñ), ( ) where j is the phase shift to be introduced by the gate if both the target and control qubits are in the logical state 1ñ as requested by the gate s definition (see equation (1)). The program qubit is destroyed by detection in the process while the target and control qubits leave the gate and can be used for further processing. The paper is organized as follows: In section we derive the basic functioning of the gate. In section 3 we show what techniques can be used to increase the success probability of Figure. Linear-optical setup for the C-phase gate with programmable phase shift. The target, control, and program qubit enter the setup at T IN,C IN, and P IN, respectively and the target and control output are denoted as follows: T OUT and C OUT. The program qubit is detected by polarization-sensitive detector D by projecting it onto diagonally polarized state. PBS x (x = 1,, 3) transmit horizontally polarized photons while reflecting vertical polarization. PPBS has unit transmissivity for horizontal polarization and tv = 1 3 for vertical polarization (therefore the reflectivity for vertical polarization is rv = 3). Filter F 1 is a neutral density filter with amplitude 1 transmissivity t F1 = while the filter F only filters horizontal polarization with transmissivity tfh = 1 3. Half-wave plates (HWP) (HWP) are rotated with respect to horizontal polarization by 9., HWP and HWP The gate succeeds if one photon leaves by the target output port, one photon by the control output port and one photon is detected by detector D. the scheme. Finally, in section 4, we discuss the scheme s experimental feasibility.. Linear-optical scheme A linear-optical scheme for a C-phase gate with programmable phase shift is depicted in figure. In this scheme, we consider encoding logical states 0ñ and 1ñ into horizontal (H) and vertical (V) polarization states of individual photons. We also introduce an auxiliary mode that is similar to the Lanyon et al gate [13], but we also introduce an auxiliary mode. In our case, however, we use this mode for interaction between the target and program qubits. In this section, we describe the principle of operation of the gate for all four basis states defined in equation (1) and an arbitrary phase shift j set in the program qubit (). The gate is necessarily probabilistic (all linear-optical C-phase gates are [14]). Its successful operation is demonstrated by the observation of one photon at each of the target and control output ports, and also by the detection of a photon on detector D. Let us start with the evaluation of the state 00y P ñ (we maintain the order of qubits: target, control, and program). The target photon impinges on the polarizing beam splitter (PBS 1 ) that sends it to the upper path. There the target photon is subjected to a neutral density filter 1 F 1 with amplitude transmissivity t F1 = and subsequently continues to the target output port by passing through the second polarizing beam splitter PBS. So far, one can write S-48

123 J. Opt. 17 (015) 150 K Lemr et al down the transformation of the state as 00y P 1 00 yp. Meanwhile the control photon is transmitted by the partially polarizing beam splitter (PPBS) (with transmissivity th = 1 for horizontal polarization and tv = 1 - rv = 1 for 3 vertical polarization) and after being subjected to polarization filtering by the filter F (filtering horizontal polarization with 1 transmissivity t FH = and letting the vertical polarization 3 unfiltered tf V= 1) it leaves the setup by the control output port. At this point the transformation by the gate reads: 00y P yp. Finally, the program photon impinges on PBS 3, where it gets transmitted and reflected with equal amplitude 1. Since the gate only succeeds if a photon is detected on detector D, it is only necessary to take into account the transmission of the program photon through PBS 3. Considering that the program photon is in the state (), the overall state gets transformed into 1 00yP ñ. Once the program photon leaves PBS 3, we project it onto 1 diagonal polarization Dñ= ( 0ñ+ 1ñ, ) resulting in the final form of the transformation: 1 00yPñ D ñ. ( 3 ) This projection is needed to erase the which-path information about the photon detected on detector D. In the same way, we now evaluate the transformation of the second state 01 ypñ. The only difference this time is in the control qubit. It impinges on the PPBS having vertical polarization and is therefore transmitted with amplitude 1 3 and reflected with amplitude. Only the transmission of the 3 control photon by the PPBS contributes to the successful operation of the gate. Furthermore, no attenuation of the vertically polarized control photon takes place on F. By using the same transformation for the target and program qubits as in the previous paragraph, one can identify the overall action of the gate 1 01yP D ñ. ( 4 ) and thus causes the overall state to get transformed into yp yp + 10 yp. At this point, the target and control photons interact on the PPBS. Since the control photon is transmitted through the PPBS (having horizontal polarization in this case), we only take into account the transmission of the target photon to ensure successful outcome of the gate yp yp yp, where we have already incorporated the action of the polarization-sensitive filter F. Now the target state enters a Hadamard transform implemented by an HWP rotated by.5 with respect to horizontal polarization, providing target photon transformation of the form of 0ñ 1 ( 0ñ + 1ñ) 1ñ 1 ( 0ñ - 1 ñ), ( 6) which then translates into the overall state evolution 10y P yp. The target photon passes through PBS 3 having horizontal polarization. Thus, as in the cases derived above, the gate can only succeed if the program photon passes through the PBS 3 and then gets projected onto diagonal polarization. Thus we obtain the transformation in the form of 1 10yP D 6 00 ñ. Finally, the target photon is again subjected to a Hadamard transform (HWP 3 ), resulting in 1 10yP ( 00Dñ + 10 Dñ). 4 3 Only the target photon reflected by the PBS leaves the gate, by designated output port, and thus we obtain the final form of the transformation: 1 10yP D ñ. ( 7 ) The third state 10y P ñ is a different matter. The target photon is reflected by PBS 1 entering the lower path, where it is subjected to an HWP 1 oriented by 9. with respect to horizontal polarization. This wave plate transform the target photon in the following way: ñ ñ+ 1ñ ( 5) To complete our analysis, we now evaluate the transformation of the last basis state 11 ypñ. As in the previous case, the target photon gets reflected on PBS 1 and transformed by the HWP 1 according to equation (5). The overall state thus takes the form of yp yp + 11 yp. 3 S-49

124 J. Opt. 17 (015) 150 K Lemr et al At this point, two-photon interference on the PPBS takes place resulting in 1 11yP ( 01yP + 11yP - 11yP ) 3 1 = 01yP - 11 yp, 8 3 ( ) ( ) where only the terms contributing to success of the gate are shown. Note that when both the target and control photons enter the PPBS in vertical polarization state, the interference of both the photons being transmitted and the photons being reflected (Hong Ou Mandel interference) occurs, which introduces a phase shift π to the term 11y P ñ [10]. By means of the subsequent Hadamard transform in the target mode (HWP ), the state transforms into 11y P yp. On PBS 3 the target photon gets reflected (being vertically polarized) and so only the program photon reflection can contribute to the gate s successful operation. This means that only its vertical polarization term contributes yielding the overall state in the form of 1 11yP -ei j ñ which, after projecting the photon in program mode onto diagonal polarization, gives 1 11yP -ei j D 6 11 ñ. Action of the Hadamard gate in the target mode (HWP 3 ) and reflection of the target photon on PBS to its output port results in the final transformation: 1 11yP ei j D ñ. ( 9 ) In contrast to the three previous cases, the state is now phaseshifted by angle j exactly as prescribed in (1). We have shown that the setup depicted in figure implements the tunable C-phase gate with the phase shift programmed by the program qubit. The overall action can be summarized by the following formula: 1 xyyp xyd 4 3 e ixyj ñ, ( 10 ) where x, y = 0, 1. This has been demonstrated on all four basis states of the control and target qubits together with an arbitrary program state. Since all transformations are linear, the entire operation holds also for any superposition of the these four basis states. The success probability of the gate is ( ) = and is state-independent in order to avoid 48 deforming the superpositions of basis states. In the next section, we will consider potential improvements to the setup in order to increase the success probability. Figure 3. Optimized setup for the programmable C-phase gate. The components are designated as in figure with the newly added HWP 4 (rotated by.5 ) and HWP 5 (rotated by 45 ) and a PLM implementing a feed forward operation. 3. Increasing the success probability We have discussed the basic scheme for the programmable C-phase gate, which is the simplest to implement in the laboratory. On the other hand, its success probability is more than four times lower then the success probability of the optimal non-programmable C-phase gate. In this section, we will discuss two optimization approaches that will result in considerable improvement in success probability (figure 3). For example, one can increase the success probability by extending the set of projections used for detecting the program photon. As derived in the previous section, the gate succeeds if the program photon is projected onto diagonal polarization and detected by D. This way, we neglect half of the cases corresponding to the program photon being projected onto anti-diagonal polarization [ Añ= ( 0ñ- 1ñ]. ) 1 As experimentally demonstrated on a simpler unconditional programmable gate [18], it is possible increase the success probability by a factor of two if the anti-diagonal projections of the program photon are included. In such cases, a feedforward transformation 1ñ - 1ñhas to be applied to the target photon immediately as it exits PBS 3 [4]. Such a transformation can be achieved by using, for instance, a phase modulator (PLM) [18]. Alternatively, one can also increase the overall success probability by a factor of two by using both the output ports of PBS (designated T OUT1 and T OUT in figure 3). In this case, the amplitude transmissivity of the filter F 1 shall be 1 reduced to t F1 = and an HWP 4 inserted behind it. This newly added wave-plate is rotated by.5 with respect to horizontal polarization to implement the Hadamard transform (6). The target state at T OUT1 is thus kept unchanged, but it allows for the target photon to leave also by the output port T OUT. The target photon exiting PBS by its second output is however polarization-swapped with respect to the target photon in the first output. An HWP 5 (rotated by 45 ) is therefore inserted to the output port T OUT to perform the 4 S-50

125 J. Opt. 17 (015) 150 K Lemr et al swap operation ( 0ñ 1 ñ, 1ñ 0ñ). The second output port can be used only if one does not require the target photon to leave by a specified output. Such a situation occurs, for instance, if the target qubit is immediately measured after being processed by the gate (the gate is the last element in a quantum circuit). The measurement apparatus can then be installed to both output ports. If only one of the mentioned optimization strategies is used, the success probability of the gate increases to 1/4, and if both of them are used, it reaches the value of 1/1 (for all phases j). Note that optimal non-programmable C-phase gate has a minimal success probability about 1/11 for j close to p [15]. If completely optimized, the programmable gate thus 3 performs with almost the same probability as the optimal nonprogrammable gate at its success probability minimum. 4. Experimental feasibility In this section, we discuss the feasibility of the proposed scheme based on the current level of technological development in linear-optical quantum information processing with discrete photons. First, in order to achieve any linear-optical quantum gate, one needs to generate an adequate input photon state. Our gate requires three photons, each bearing one polarization-encoded qubit. The generation of three separate photons has already been achieved in various experiments. Either one photon pair from spontaneous parametric downconversion (SPDC) is combined with one additional photon from attenuated fundamental laser beam [5] or two photon pairs are generated via SPDC with one photon serving just as a trigger [0]. With either of these techniques, one can generate input states with sufficient fidelity, typically more than 95%. Repetition rate at the output of experimental setup is typically Hz [6, 7]. In the next step, we asses the feasibility of stabilizing the proposed scheme. There are two types of stabilizations required: two-photon temporal and spatial overlap stabilization (Hong Ou Mandel interference [8]) and single-photon interference stabilization. The overlap stabilization requires the precision of about 1/50 of the photon wave packet full width at half maximum (FWHM) which is typically 100 μm. One can use motorized translation to achieve this task and the adjustment remains stable for hours. On the other hand, the single-photon interference typically needs to be stabilized to at least l 50, where λ is the photon s wavelength. Such precision requires the combination of both motorized translation for larger steps and piezo translation for fine adjustments. In a typical bulk interferometer on decimeter scale, the single-photon stability lasts for less than a minute [15]. One can significantly increase the duration by replacing a classical interferometer with a more compact design [9]. Finally, the feasibility considerations have to be dedicated to the final detection procedure. The detection has to be robust against non-unit quantum detection efficiency of typical detectors and technological losses (e.g., back-reflection and coupling efficiency). This requirement rules out vacuum detection-based schemes (schemes where success is demonstrated by vacuum detection or no detection) [30]. Similarly, photon-number resolving detection is not completely reliable because of detection (in)efficiency [31] and technological losses. In our case, however, the scheme only requires post-selection on three-fold coincidence detections, thus, the quantum efficiency of the detectors only affects the detection rate and not the detected quantum state. This is yet another key feature that contributes to the feasibility of the proposed scheme. It should be noted that our scheme can also be implemented on the platform of integrated optics. In contrast to the traditional bulk-optical platform, integrated optics brings many interesting benefits including the inherent stability of interferometers and the possible usage in practical quantum information-processing devices. Recent developments in this field makes us confident that the proposed scheme is well within reach. For instance, single-photon and two-photon interferometers with adjustable phase shifts have been achieved experimentally [7]. Simultaneously, investigators have already demonstrated complete control over polarization-encoded qubits in integrated optical circuits [3, 33]. 5. Conclusion In this paper, we have provided a linear-optical scheme for a programmable C-phase gate. The phase shift introduced by this gate is set by the state of a program qubit, which makes the gate more versatile than previously implemented tunable C-phase gates with phase shift hard-coded to the equipment setting. The setup is designed with experimental feasibility in mind. It does not require photon-number resolving detectors or post-selection on vacuum detection. Therefore, it is feasible given the current understanding of experimental linear-optical quantum information processing. We have also presented two optimization options. Each of them doubles the overall success probability of the gate. Using both of these optimizations, the gate succeeds with probability of 1/1, which is close to the success probability of a non-programmable tunable C-phase gate [15]. Furthermore, the two optimization steps can be used independently. If only one of them is used, the success probability equals 1/4. The first optimization method consists of applying an experimentally feasible feed-forward operation. The second optimization method involves using both output ports of the final polarizing beam splitter in the target mode. Acknowledgments The authors thank Jára Cimrman for his helpful suggestions. K L acknowledges support by the Czech Science Foundation (Grant No P). K B acknowledges support by the Foundation for Polish Science and the Polish National Science Centre under grant No. DEC-011/03/B/ST/ Finally, the authors acknowledge the project No. LO1305 of the Ministry of Education, Youth and Sports of the Czech Republic. 5 S-51

126 J. Opt. 17 (015) 150 K Lemr et al References [1] Neilsen M A and Chuang I L 00 Quantum Computation and Quantum Information (Boston: Cambridge University Press) [] Bouwmeester D, Ekert A and Zeilinger A 001 The Physics of Quantum Information (Heidelberg: Springer) [3] Barenco A, Bennett C H, Cleve R, DiVincenzo D P, Margolus N, Shor P, Sleator T, Smolin J A and Weinfurter H 1995 Elementary gates for quantum computation Phys. Rev. A [4] Shende V V, Markov I L and Bullock S S 004 Minimal universal two-qubit controlled-not-based circuits Phys. Rev. A [5] Cory D, Price M, Fahmy A and Havel T 1997 Ensemble quantum computing by NMR spectroscopy Proc. Natl Acad. Sci. USA [6] Schmidt-Kaler F, Haffner H, Riebe M, Gulde S, Lancaster G P T, Deuschle T, Becher C, Roos C F, Eschner J and Blatt R 003 Realization of the Cirac Zoller controlled NOT quantum gate 003 Nature [7] Plantenberg J H, de Groot P C, Harmans C J P M and Mooij J E 007 Demonstration of controlled-not quantum gates on a pair of superconducting quantum bits Nature [8] Ralph T C, Langford N K, Bell T B and White A G 00 Linear optical controlled-not gate in the coincidence basis Phys. Rev. A [9] O Brien J L, Pryde G J, White A G, Ralph T C and Branning D 003 Demonstration of an all-optical quantum controlled- NOT gate Nature [10] Kiesel N, Schmid C, Weber J, Ursin R and Weinfurter H 005 Linear optics controlled-phase gate made simple Phys. Rev. Lett [11] Kok P, Munro W J, Nemoto K, Ralph T C, Dowling J P and Milburn G J 007 Linear optical quantum computing with photonic qubits Rev. Mod. Phys [1] Bartkowiak M and Miranowicz A 010 Linear-optical implementations of the iswap and controlled NOT gates based on conventional detectors J. Opt. Soc. Am. B [13] Lanyon B P, Barbieri M, Almeida M P, Jennewein T, Ralph T C, Resch K J, Pryde G J, O Brien J L, Gilchrist A and White A G 009 Simplifying quantum logic using higher dimensional Hilbert spaces Nat. Phys [14] Kieling K, O Brien J L and Eisert J 010 On photonic controlled phase gates New J. Phys [15] Lemr K, Cernoch A, Soubusta J, Kieling K, Eisert J and Dušek 011 Experimental implementation of the optimal linear-optical controlled phase gate Phys. Rev. Lett [16] Nielen M A and Chuang I L 1997 Programmable quantum gate arrays Phys. Rev. Lett Vidal G, Masanes L and Cirac J I 00 Storing quantum dynamics in quantum states: a stochastic programmable gate Phys. Rev. Lett Hillery M, Bužek V and Ziman M 00 Probabilistic implementation of universal quantum processors Phys. Rev. A [17] Mičuda M, Ječek M, Dušek M and Fiurášek J 008 Experimental realization of a programmable quantum gate Phys. Rev. A [18] Miková M, Fikerová H, Straka I, Mičuda M, Fiurášek J, Ježek M and Dušek M 01 Increasing efficiency of a linear-optical quantum gate using electronic feed-forward Phys. Rev. A [19] Lemr K, Bartkiewicz K, Černoch A and Soubusta J 013 Resource-efficient linear-optical quantum router Phys. Rev. A [0] Vitelli Ch, Spagnolo N, Aparo L, Sciarrino F, Santamato E and Marrucci L 013 Joining the quantum state of two photons into one Nat. Photon [1] Franson J D, Donegan M M and Jacobs B C 004 Generation of entangled ancilla states for use in linear optics quantum computing Phys. Rev. A [] Lemr K 011 Preparation of Knill Laflamme Milburn states using tunable controlled phase gate J. Phys. B: At. Mol. Opt. Phys [3] Lemr K, Bartkiewicz K, Černoch A and Soubusta J 015 Experimental implementation of optimal linear-optical controlled-unitary gates Phys. Rev. Lett [4] Bula M, Bartkiewicz K, Černoch A and Lemr K A 013 Entanglement-assisted scheme for nondemolition detection of the presence of a single photon Phys. Rev. A [5] Slodička L, Ježek M and Fiurášek J 009 Experimental demonstration of a teleportation-based programmable quantum gate Phys. Rev. A (R) [6] Dobek K, Karpinski M, Demkowicz-Dobrzanski R, Banaszek K and Horodecki P 013 Experimental generation of complex noisy photonic entanglement Laser Phys [7] Metcalf B J et al 013 Multiphoton quantum interference in a multiport integrated photonic device Nat. Commun [8] Hong C K, Ou Z Y and Mandel L 1987 Measurement of subpicosecond time intervals between two photons by interference Phys. Rev. Lett [9] Mičuda M, Sedlák M, Straka I, Miková M, Dušek M, Ježek M and Fiurášek J 013 Efficient experimental estimation of fidelity of linear optical quantum toffoli gate Phys. Rev. Lett [30] VanMeter N M, Lougovski P, Uskov D B, Kieling K, Eisert J and Dowling J P 007 General linear-optical quantum state generation scheme: applications to maximally path-entangled states 007 Phys. Rev. A [31] Rěhaćěk J, Hradil Z, Haderka O, Peřina J Jr and Hamar M 003 Multiple-photon resolving fiber-loop detector Phys. Rev. A (R) [3] Crespi A et al 011 Integrated photonic quantum gates for polarization qubits Nat. Commun. 566 [33] Corrielli et al 014 Rotated waveplates in integrated waveguide optics Nat. Commun S-5

127 PHYSICAL REVIEW A 88, 0137 (013) Entanglement-based linear-optical qubit amplifier Evan Meyer-Scott, 1 Marek Bula, Karol Bartkiewicz,,* Antonín Černoch, 3 Jan Soubusta, 3 Thomas Jennewein, 1 and Karel Lemr 3, 1 Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, 00 University Avenue W, Waterloo, Ontario, Canada NL 3G1 RCPTM, Joint Laboratory of Optics of Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic, 17. listopadu 1, Olomouc, Czech Republic 3 Institute of Physics of Academy of Sciences of the Czech Republic, Joint Laboratory of Optics of PU and IP AS CR, 17. listopadu 50A, Olomouc, Czech Republic (Received 6 June 013; published 6 July 013) We propose a linear-optical scheme for an efficient amplification of a photonic qubit based on interaction of the signal mode with a pair of entangled ancillae. In contrast to a previous proposal for qubit amplifier by Gisin et al. [Phys Rev. Lett. 105, (010)], the success probability of our device does not decrease asymptotically to zero with increasing gain. Moreover, we show how the device can be used to restore entanglement deteriorated by transmission over a lossy channel and calculate the secure key rate for device-independent quantum key distribution. DOI: /PhysRevA PACS number(s): Hk, 4.50.Dv, Lx I. INTRODUCTION The fundamentals of quantum physics were discovered and formulated nearly a hundred years ago. Three decades ago scientists postulated that the laws of quantum physics could be used to improve capabilities of computation and communication technologies [1]. This idea sparked intense research resulting in the discovery of many quantum information protocols, some of them even with practical, modern implementations [,3]. One such application of quantum information is quantum cryptography, comprising various quantum key distribution protocols (QKD) [4]. QKD offers unconditional security of private communications certified by the laws of quantum physics. In the real world, QKD suffers from various technological limits, especially the need to trust imperfect detectors and single photon sources, quantum channel losses, and background noise. The latter effects limit the maximum distance for unconditionally secure communications [5]. Long-distance QKD has been realized over 144 km in free-space [6] and over 60 km in an optical fiber [7]. Trust in the imperfect devices used for cryptography allows eavesdroppers to attack unintended leakages of information or control detectors, known as side channels [8]. The side channel attacks can be solved in principle by using Bell-state projection measurements or using entanglementbased protocols. The simpler approach is measurementdevice-independent QKD [9 11]. In this case a projection on a Bell state in the middle of the communication line removes all detector side channels. The more complete approach is device-independent QKD (DI-QKD) [1 16] and its security is based on the loophole-free violation of a Bell inequality. DI-QKD removes all source and detector side channels but requires closing of the detector (high-efficiency detection) and * bartkiewicz@jointlab.upol.cz k.lemr@upol.cz locality (distant detectors) loopholes, which has not yet been achieved simultaneously [17]. For DI-QKD and other protocols requiring high-efficiency detection, a method is required to circumvent the channel losses inherent in photon transmission. In classical optical communication networks the problem of losses is solved using amplifiers of the classical signal. For quantum communication, losses are more fundamental. The quantum signals are stored in polarization or temporal modes of individual photons and any quantum amplifier is bound by the quantum limits like the no-cloning theorem [18]. Several proposals of quantum amplifiers were recently introduced, wherein the quantum limit can be circumvented by making the amplification nondeterministic. This type of amplification is called heralded noiseless amplification [19] and is already seeing successful implementation [1]. Note that there exists a complete equivalence between distribution of two-qubit entanglement and secure key distribution [0]. In other words, any quantum channel is capable of secret communication if and only if it is capable of distributing entanglement. In this article we propose a scheme of a linear-optical qubit amplifier that can restore the attenuated qubit and is also capable of distilling deteriorated entanglement of the qubit state. Our amplifier is ready to be used in DI-QKD schemes. Moreover, it outperforms previously published proposals. In contrast tothe Gisinet al. scheme [13], the success probability of our device does not asymptotically approach zero when increasing the amplification gain. Furthermore, in comparison to the Pitkanen et al. scheme [14], our device provides tunable gain and for the case of infinite gain allows better success probability due to its intrinsic elimination of the two-photon component after heralding. However, the Pitkanen et al. device may perform better when using a probabilistic source for the ancilla photons, due to its extra stage of heralding. The scheme by Curty and Moroder makes use of entanglement as in our device, but it is limited to infinite gain only [15], and in this regime it performs comparably to our device. Further to these works, we present a thorough investigation of the gain versus /013/88(1)/0137(7) American Physical Society S-53

128 EVAN MEYER-SCOTT et al. PHYSICAL REVIEW A 88, 0137 (013) success probability tradeoff which is a crucial figure of merit for probabilistic amplifiers. The paper is organized as follows. The principle of the amplifier operation is explained in Sec. II. The entanglement distillation is analyzed in Sec. IIIand DI-QKD is discussed in Sec. IV. Conclusions are drawn in the final Sec. V. II. PRINCIPLE OF OPERATION The amplifier (depicted in Fig. 1) consists of four polarizing beam splitters. Two of them (PBS in and PBS out ) form a Mach-Zehnder interferometer between signal input port in and output port out. These polarizing beam splitters totally transmit horizontally polarized light while totally reflect light with vertical polarization. The other two are partially polarizing beam splitters, denoted as PPBS 1 and PPBS, and placed in their respective arms of the interferometer. PPBS 1 reflects vertically polarized light, while having reflectivity r for horizontal polarization. In terms of creation operators this transformation reads â in,h râ out,h + 1 r â D1,H, â a1,h râ D1,H + 1 r â out,h, â a1,v â D1,V, where labeling of spatial modes has been adopted from Fig. 1 and H, V denote horizontal and vertical polarizations. Similarly the PPBS reflects completely the horizontal polarization and with reflectivity r it reflects vertically polarized photons. The parameter r is to be tuned as explained below. Successful operation of the amplifier is heralded by twophoton coincidence detection on detection blocks D 1 and D. To demonstrate the principle of operation, let us assume the input signal to be a coherent superposition of vacuum and a polarization-encoded single photon qubit ψ in =α 0 +β H H +β V V, where 0 denotes vacuum, H, V denote horizontal and vertical polarization states, respectively, and the coefficients meet the normalization condition α + β H + β V = 1. in a r H =1 r V = r - PBS in PPBS D PPBS 1 PBS out a 1 out - r H = r r V =1 FIG. 1. (Color online) Scheme for entanglement-based linearoptical qubit amplifier as described in the text. D 1 and D are standard polarization analysis detection blocks (for reference see [1]). D 1 The amplifier makes also use of a pair of ancillary photons impinging on ports a 1 and a of PPBS 1 and PPBS, respectively. These ancillary photons are initially in a maximally entangled Bell state of the form + a 1a = 1 ( H a1 H a + V a1 V a ), where the indices denote the ancillary photons spatial modes. The total state entering the amplifier composed of the signal and ancillary photons reads ψ T = ψ in + a 1a = 1 [α 0 in H a1 H a +α 0 in V a1 V a + β H H in H a1 H a +β H H in V a1 V a + β V V in H a1 H a +β V V in V a1 V a ]. Now we inspect evolution of all the individual terms present in the previous equation. Since the successful operation of the amplifier is conditioned by a two-photon coincidence detection by D 1 and D we postselect only such cases: 0 in H a1 H a r 0 out H D1 H D, 0 in V a1 V a r 0 out V D1 V D, H in H a1 H a (r 1) H out H D1 H D, H in V a1 V a r H out V D1 V D, V in H a1 H a r V out H D1 H D, V in V a1 V a (r 1) V out V D1 V D. Note that for r = 0 it is impossible to have more than one photon in the output mode, even for multiple photons in the input mode. Subsequently we perform polarization-sensitive detection on D 1 and D in the basis of diagonal D ( H + V ) and antidiagonal A ( H V ) linear polarization. This way we erase the information about the ancillary state and project the signal at the output port to ψ out αr 0 + 3r 1 (β H H +β V V ), where we have incorporated the fact that only if both the detected polarizations on D 1 and D are identical (DD or AA coincidences) the device heralds a successful amplification and thus only one half of the measurement outcomes contributes to success probability. At this point we define the amplification gain G as a fraction between signal and vacuum probabilities G = (3r 1) (1) 4r and calculate the corresponding success probability P = r [ α + G( β H + β V )]. () Note that while the gain itself is input state independent, the success probability depends on both the gain and the input state parameters. This reflects the intuitive fact that it is for instance impossible to amplify a qubit that is actually not present in the input state (β H = β V = 0). Let us analyze the results further. As expected the gain G = 1 is obtained for r = 1 with success probability P = S-54

129 ENTANGLEMENT-BASED LINEAR-OPTICAL QUBIT AMPLIFIER PHYSICAL REVIEW A 88, 0137 (013) Success probability Gain = 0.30 = 0.50 = 0.95 G nom ( = 0.95) Gisin ( = 0.50) FIG.. (Color online) Success probability is depicted as a function of gain for three different input states parametrized by α. For comparison, the success probability of the Gisin et al. scheme [13] is presented (in this case α = 0.5). Note that the success probability of our amplifier converges asymptotically to a nonzero value for any state with α 1. Success probability is also plotted as a function of nominal gain G nom for the case of α = Note that according to its definition (3), the nominal gain is upper bounded by the value of 0 in this particular case (blue X symbol). independent on the input state. On the other hand, an infinite gain is obtained for r = 0 with success probability of P = ( β H + β V )/4. In this particular case, it is however possible to increase the success probability twice by including also detection coincidences DA and AD accompanied by a feed-forward operation V V on the output state. Note that this regime is suitable for nondemolition presence detection of the qubit []. Figure depicts the tradeoff between success probability and gain for three different input states containing different amounts of vacuum. In a recent paper [1], its authors proposed also another measure of amplifier performance the nominal gain G nom defined as G G nom α + G( β H + β V ) = r G P. (3) While the ordinary gain G describes how much the qubit to vacuum intensity ratio has been increased under the amplification procedure, the nominal gain shows how much the overall success probability of finding the qubit state has increased. For this reason, the nominal gain is bound by the inverse value of the initial qubit probability (e.g., for β H + β V = 0., the maximum value of nominal gain is 5 and in this case the vacuum state is completely eliminated). Figure depicts the success probability as a function of nominal gain for one particular initial state ( α = 0.95). It is worth noting that in contrast to the Gisin et al. scheme [13], the success probability does not decrease asymptotically to 0 with increasing gain (also illustrated in Fig. for comparison). One may however suggest that in the case of infinite gain, the scheme performs exactly as well as standard teleportation. While this is indeed true, standard teleportation does not allow us to tune the amplification gain and therefore the superposition of vacuum and qubit state collapses either onto vacuum or qubit state. In contrast, our scheme allows for the coherent superposition of these two terms to be maintained. Keeping coherence between vacuum and qubit terms is crucial for instance in all applications involving dual rail encoding. III. AMPLIFICATION-BASED ENTANGLEMENT DISTILLATION Quantum entanglement is one of the key ingredients in quantum communications. It can be used for teleportation [3], quantum cryptography [4], or remote state preparation [5]. It is also very sensitive to losses and decoherence occurring in the communication channel [6 8]. For this reason, entanglement distillation the way of improving entanglement of a state subjected to some degradation is a very important tool in quantum communications [9,30]. In this section we show how the amplifier can be used to distill entanglement on an example entangled state in dual-rail encoding. Suppose an unknown polarization qubit ψ is distributed in two spatial modes creating thus maximally entangled state of the form = 1 ( ψ0 + 0ψ ). (4) States of vacuum and qubit superposition are needed in various quantum communication protocols (e.g., quantum secret sharing [31]) and are indispensable in implementations combining spatial and polarization encoding [3 34]. Now let us consider a lossy channel with transmissivity 1 T>0 used to distribute the second spatial mode of this entangled state. This channel would deteriorate the state to ˆρ(α,p) = (1 p) p α α, where α = α 0ψ + 1 α ψ0, with α = T/(T + 1) and p = (T + 1)/. This state belongs to the class of amplitude damped states from Ref. [8] where the entanglement and nonlocality of such states was studied. Since various measures of entanglement have different operational meaning, below we consider amplification of a few popular entanglement measures analyzed in [8] (for a review on entanglement measures see [35]). The negativity (concurrence) of the mixed state before amplification is simply N = T (C = T ). After the amplification in the lossy mode the parameters of the state ˆρ(α,p) read α = GT /(GT + 1) and p = N (GT + 1)/, where G denotes the gain as defined in the previous section and N = /( + GT T ). The entanglement of ˆρ(α,p) (seeref.[8]) can be quantified by its concurrence C = p α(1 α) = N GT, which can be further used to express its negativity as N = 1 [ (1 p) + C (1 p)] = N [ (1 T ) + 4GT (1 T )]. The third prominent measure of entanglement is the relative entropy of entanglement S, but as demonstrated by Miranowicz and Ishizaka [36] finding a closed formula for S in case of the amplitude-damped states requires solving a single variable S-55

130 EVAN MEYER-SCOTT et al. PHYSICAL REVIEW A 88, 0137 (013) Negativity T = 0.50 T = 0.5 T = 0.10 T = Gain FIG. 3. (Color online) Negativity of entanglement depicted as a function of amplification gain for several different channel transmissivities T. A maximally entangled state formed of superposition of vacuum and qubit state is subjected to a channel with transmissivity T resulting in entanglement loss. Suitably set amplification gain can increase the amount of entanglement. The wide gray curve joins the maxima of negativity for all values of transmissivity T and subsequent optimal gains. equation for which no general analytic solution is known. Hence we calculate S numerically as described in [8,36]. As shown on the example of negativity in Fig. 3 the entanglement measures are functions both of transmissivity T and gain G. The optimal gain for maximizing the entanglement is G opt,n = 1 T [ T T (T 1)] for negativity and G opt,c = ( T )/T for concurrence. We do not present the exact expression for S and its optimal gain, but the G opt,s curve obtained numerically is presented together with other G opt curves in Fig. 4. The curves shown in Fig. 4 do not overlap, thus the optimal gain G opt varies depending on the entanglement measure to be used. However, Fig. 4 suggests that for any value of T>0, there is an optimal gain G opt 1 regardless of the applied entanglement measure. The T G opt G opt, C G opt, N G opt, S 1/T T FIG. 4. (Color online) The optimal gain G opt for various entanglement measures as function of channel transmissivity T. Setting the optimal gain allows to obtain the largest possible value of the selected entanglement measure for a given loss parametrized by T. Entanglement C C opt N N opt S S opt T FIG. 5. (Color online) The entanglement measures before (C, N, S) andafter(c opt,n opt,s opt ) optimal amplification as functions of channel transmissivity T. entanglement measures before and after optimal amplification are depicted in Fig. 5 as functions of T. Note that for gain reaching infinity (standard teleportation), the entangled state would collapse onto the qubit state thus destroying the entanglement. The corresponding success probability of the amplification process is P succ = r N = G G + 3G + 3, 9N where r follows from Eq. (1). InFig.3 we plot the amplified negativity as a function of the chosen gain for several different values of channel transmissivity. Note that our results for negativity, especially the expression for optimal gain G opt,n, are also valid for logarithmic negativity log (N + 1) which is a concave function of N providing an upper bound to the distillable entanglement [37,38] given that the state was predistilled using the above-described procedure. The above-performed calculations reveal how qubit amplification can be used for partial entanglement recovery. However in neither of the cases, the entanglement has been restored to the original maximum value due to the presence of the vacuum term Recently, Mičuda et al. experimentally demonstrated a rather clever way to eliminate the presence of such a term [39]. They considered only vacuum and a fixed polarization single photon state, but the technique can be adopted for qubit amplification as well. Their approach is based on deliberate coherent attenuation before the state is transmitted via the lossy channel. This coherent attenuation is performed by subjecting the state to a beam splitter with transmissivity ν and subsequent postselection on vacuum in the ancillary mode. With the probability of ν, one can thus disbalance the original state (4) to α, where α = ν/(ν + 1). The choice of attenuation factor ν influences the probability p = (1 νt )/(1 + ν) and α = νt/(1 νt )in the density matrix ˆρ(α,p) of the state α transmitted through the lossy channel. Subsequent amplification will increase α thus also the entanglement of the state. Ideally for ν 0 and gain G the original negativity can be completely restored. Of course such parameters lead to zero success rate so there is a need for some sort of compromise. Nevertheless this S-56

131 ENTANGLEMENT-BASED LINEAR-OPTICAL QUBIT AMPLIFIER PHYSICAL REVIEW A 88, 0137 (013) Negativity T = 0.50 T = T = Success probability FIG. 6. (Color online) Negativity and success probability tradeoff obtained using coherent attenuation before transferring the state through a lossy channel. This tradeoff is depicted for three different values of channel transmissivity T. Even though this strategy allows us to increase the negativity arbitrarily close to 1, the product of negativity and success probability is maximized when no coherent attenuation is used. line of reasoning demonstrates the importance of amplification with high gain, where our amplifier outperforms the original Gisin et al. proposal [13]. The above mentioned compromise can be quantified using the entangling efficiency E eff of the protocol [40]. The entangling efficiency is an entanglement generation measure suitable for probabilistic devices. In contrast to a more widely used entangling power [41 43], the entangling efficiency optimizes over the device parameters in order to maximize the product of success probability and negativity (or any other entanglement measure) E eff = max{p succ N}. The negativity is calculated similarly as presented above using the analytical form of the density matrix. The success probability is composed of the success probability of attenuation (ν) and the success probability of amplification [Eq. ()]. In order to find the best strategy, we perform a numerical simulation. The plot in Fig. 6 shows the tradeoff between negativity and success probability obtained when using the coherent attenuation strategy. This simulation also reveals that the product of success probability and negativity is maximized for ν = 1 in all cases. So as far as the entanglement rate described by the entangling efficiency is concerned, the coherent attenuation does not offer any improvement. On the other hand, it is important to note that this strategy finds its merit when the goal is to achieve high negativity or high fidelity at the output. IV. DEVICE-INDEPENDENT QUANTUM KEY DISTRIBUTION Photon amplifiers can find additional applications in device-independent quantum key distribution, a stronger form of entanglement-based quantum cryptography based on the violation of Bell s inequality [16]. As mentioned above, DI-QKD does not require any knowledge of Alice and Bob s measurement devices, but does require closing the detection loophole [44]. A number of ways of closing this loophole have been demonstrated, including using trapped ions [45,46] and efficient photon detection [17], but none has done so over the long distances needed for cryptography due to the intrinsic loss associated with photon transmission in fiber or free space. Gisinet al. recently proposed using a photon amplifier to herald incoming photons, closing the detection loophole and allowing DI-QKD [13]. In their scheme, as in the recently proposed improvements [14,15], a source of photons near Alice emits maximally entangled photon pairs. One photon is sent to Alice, which she detects directly with high efficiency, and the other photon is sent over a long channel to Bob. Bob routes the incoming photon through some heralded amplifier (e.g., the one proposed by Gisin et al. or by us) before detection, closing the detection loophole by performing a Bell measurement only upon successful amplification. In order to compare the performance of the three previous amplifiers with ours, we performed numerical quantum optical simulations of the amplifiers. The initial source of entanglement was spontaneous-parametric down-conversion, with photon pair probability set to 10 3, and both amplifiers used on-demand photon sources (two single photons for the Gisin et al. and Pitkanen et al. schemes and a maximally entangled Bell state for ours) as ancillae. To mirror a likely experimental scenario, we used bucket detectors with 95% detection efficiency and 91% coupling efficiency as herald-ing detectors, and untrusted noiseless photonnumber resolving detectors with the same efficiency for the detection of the photons for the Bell test after heralding. The former are modeled on fast superconducting nanowire detectors [47] and the latter transition edge sensors [48]. We optimized all amplifiers over their tunable beam splitter reflectivity at each point. Finally we calculated the secure key rate per laser pulse from Eq. (11) of the Supplementary Material of Ref. [13] R = μ cc [1 h(q) I E (S,μ)], (5) where μ cc is the probability of a conclusive event for both Alice and Bob, h(q) is the binary entropy function of the measured quantum bit error rate, and I E (S,μ) is Eve s information based on the Bell inequality violation S and the ratio of inconclusive to conclusive results μ (see Eq. (3) of Ref. [13] for the full expression). As shown in Fig. 7, our amplifier outperforms the Gisin et al. scheme and can also tolerate more dark counts in the heralding detectors. This is because high gain is required to close the detection loophole after a lossy channel, and, as seen above, the success probability of the Gisin et al. photon amplifier converges asymptotically to zero for high gain. It additionally outperforms the Pitkanen et al. scheme by a nearly constant factor, where this factor comes from improvements in success probability and the ratio of conclusive to inconclusive events after heralding. This is possible because in the Pitkanen et al. scheme, the elimination of the unwanted two-photon component even for ideal ancilla photons after heralding comes at the cost of vanishing success probability, a tradeoff our amplifier does not suffer from. The optimal key rate in this DI-QKD scenario for our amplifier occurs with r = 0 for all values of channel loss, such that it performs identically to the Curty and Moroder proposal [15]. However, there could be a regime (e.g., with noise in the final Bell test detectors) where S-57

132 EVAN MEYER-SCOTT et al. PHYSICAL REVIEW A 88, 0137 (013) et al. et al. FIG. 7. (Color online) Key rate per laser pulse for deviceindependent quantum key distribution versus Bob s channel loss and dark counts per second in heralding detectors. Assuming 100 ps timing resolution in the heralding detectors leads to and 10 8 dark count probability per pulse for 1 and 100 dark count/s, respectively. Our entangled photon amplifier allows more key to be extracted than the Gisin et al. scheme, and even shows better scaling with loss. It additionally delivers approximately 1 times the key rate of the Pitkanen et al. scheme. higher success probability is needed to maximize key rate, at the cost of a larger vacuum component after the amplifier. V. CONCLUSION In this paper we have presented a linear-optical qubit amplifier. With the help of a maximally entangled photon pair, this device is able to change the ratio between vacuum and single qubit component, thus introducing qubit gain. In contrast to other proposals, our scheme achieves infinite gain with nonzero probability of success. Moreover, we have shown that the success probability of implementing infinite gain equals to the success probability of standard teleportation. To demonstrate the capabilities of our amplifier, we have presented two of its potential applications: entanglement distillation and quantum key distribution. First, the analysis of entanglement distillation reveals that our amplifier can at least partially improve entanglement deteriorated by lossy transmission. We have presented the calculation of optimal gain for three different measures of entanglement (negativity, concurrence, and relative entropy of entanglement) as a function of channel attenuation. Second, for device-independent quantum key distribution we have presented the significant improvement made by this amplifier over the previously proposed devices, including a key rate more than three orders of magnitude better for 100 km transmission distance. Practical implementation of the proposed scheme will be limited by available technology such as precision of optical components, detection efficiency, and delivery efficiency of ancillae. ACKNOWLEDGMENTS The authors gratefully acknowledge the support by the Operational Program Research and Development for Innovations European Regional Development Fund (Project No. CZ.1.05/.1.00/ ). A.Č. acknowledges Project No. P05/1/038 of Czech Science Foundation. K.B. and K.L. acknowledge support by Grant No. DEC-011/03/B/ST/01903 of the Polish National Science Centre and K.B. also by the Operational Program Education for Competitiveness European Social Fund Project No. CZ.1.07/.3.00/ M.B. acknowledges the financial support by Internal Grant Agency of Palacký University (No. PrF_013_006). E.M.S. and T.J. acknowledge support from the Natural Sciences and Engineering Research Council of Canada. K.L. also acknowledges the support by the Czech Science Foundation (Grant No P). The authors thank Norbert Lütkenhaus for helpful suggestions. [1] S. J. Wiesner, SIGACT News 15, 78 (1983). [] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 00). [3] D. Bruß and G. Leuchs, Lectures on Quantum Information (Wiley-VCH, Berlin, 006). [4] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (00). [5] K. Bartkiewicz, K. Lemr, A. Černoch, J. Soubusta, and A. Miranowicz, Phys.Rev.Lett.110, (013). [6] R. Ursin et al., Nat. Phys. 3, 481 (007). [7] S. Wang et al., Opt. Lett. 37, 1008 (01). [8] L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, Nat. Photon. 4, 686 (010). [9] H.-K. Lo, M. Curty, and B. Qi, Phys. Rev. Lett. 108, (01). [10] P. Chan, J. A. Slater, I. Lucio-Martinez, A. Rubenok, and W. Tittel, arxiv: v3. [11] Yang Liu et al.,arxiv: v1. [1] S. Kocsis, G. Y. Xiang, T. C. Ralph, and G. J. Pryde, Nat. Phys. 9, 3 (013). [13] N. Gisin, S. Pironio, and N. Sangouard, Phys. Rev. Lett. 105, (010). [14] D. Pitkanen, X. Ma, R. Wickert, P. van Loock, and N. Lütkenhaus, Phys. Rev. A 84, 035 (011). [15] M. Curty and T. Moroder, Phys. Rev. A 84, (R) (011). [16] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Phys.Rev.Lett.98, (007). [17] M. Giustina, A. Mech, S. Ramelow, B. Wittmann, J. Kofler, J.Beyer,A.Lita,B.Calkins,T.Gerrits,S.W.Nam,R.Ursin, and A. Zeilinger, Nature (London) 497, 7 (013). [18] W. K. Wootters and W. H. Zurek, Nature (London) 99, 80 (198). [19] G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, Nat. Photon. 4, 316 (010). [0] A. Acín, L. Masanes, and N. Gisin, Phys. Rev. Lett. 91, (003). [1] E. Halenková, A. Černoch, K. Lemr, J. Soubusta, and S. Drusová, Appl. Opt. 51, 474 (01). [] M. Bula, K. Bartkiewicz, A. Černoch, and K. Lemr, Phys. Rev. A 87, (013) S-58

133 ENTANGLEMENT-BASED LINEAR-OPTICAL QUBIT AMPLIFIER PHYSICAL REVIEW A 88, 0137 (013) [3] D. Bouwmeester, J. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Nature (London) 390, 575 (1997). [4] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991). [5] J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, Phys. Rev. Lett. 105, (010). [6] Ş. K. Özdemir, K. Bartkiewicz, Y. X. Liu, and A. Miranowicz, Phys. Rev. A 76, 0435 (007). [7] E. Halenková, K. Lemr, A. Černoch, and J. Soubusta, Phys. Rev. A 85, (01) [8] B. Horst, K. Bartkiewicz, and A. Miranowicz, Phys.Rev.A87, (013). [9] T. Yamamoto, M. Koashi, and N. Imoto, Phys.Rev.A64, (001). [30] Z. Zhao, J.-W. Pan, and M. S. Zhan, Phys.Rev.A64, (001). [31] M. Hillery, V. Bužek, and A. Berthiaume, Phys.Rev.A59, 189 (1999). [3] K. Lemr, A. Černoch, J. Soubusta, and J. Fiurášek, Phys. Rev. A 81, 0131 (010). [33] K. Lemr and A. Černoch, Opt. Commun. 300, 8 (013). [34] K. Lemr, K. Bartkiewicz, A. Černoch, and J. Soubusta, Phys. Rev. A 87, (013). [35] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (009). [36] A. Miranowicz and S. Ishizaka, Phys. Rev. A78, (008). [37] A. Peres, Phys.Rev.Lett.77, 1413 (1996). [38] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 3, 1 (1996). [39] M. Mičuda, I. Straka, M. Miková, M. Dušek, N. J. Cerf, J. Fiurášek, and M. Ježek, Phys.Rev.Lett.109, (01). [40] K. Lemr, A. Černoch, J. Soubusta, and M. Dušek, Phys.Rev.A 86, 0331 (01). [41] P. Zanardi, Ch. Zalka, and L. Faoro, Phys.Rev.A6, (R) (000). [4] M. M. Wolf, J. Eisert, and M. B. Plenio, Phys. Rev. Lett. 90, (003). [43] J. Batle, M. Casas, A. Plastino, and A. R. Plastino, Opt. Spectrosc. 99, 371 (005). [44] P. Pearle, Phys. Rev. D, 1418 (1970). [45] M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, Nature (London) 409, 791 (001). [46] D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, and C. Monroe, Phys.Rev.Lett.100, (008). [47] F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, Nat. Photon. 7, 10 (013). [48] A. E. Lita, A. J. Miller, and S. W. Nam, Opt. Express 16, 303 (008) S-59

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135 PHYSICAL REVIEW A 88, (013) State-dependent linear-optical qubit amplifier Karol Bartkiewicz, 1,,* Antonín Černoch, 3 and Karel Lemr 1, 1 RCPTM, Joint Laboratory of Optics of Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic, 17. listopadu 1, Olomouc, Czech Republic Faculty of Physics, Adam Mickiewicz University, Poznań, Poland 3 Institute of Physics of Academy of Sciences of the Czech Republic, Joint Laboratory of Optics of PU and IP AS CR, 17. listopadu 50A, Olomouc, Czech Republic (Received 18 September 013; published December 013) We propose a linear-optical setup for heralded qubit amplification with tunable output qubit fidelity. We study its success probability as a function of output qubit fidelity, showing that at the expense of lower fidelity, the setup can considerably increase probability of successful operation. These results are subsequently applied in a proposal for state-dependent qubit amplification. Similar to state-dependent quantum cloning, the apriori information about the input state allows us to optimize the qubit amplification procedure to obtain a better fidelity versus success probability trade-off. DOI: /PhysRevA PACS number(s): Hk, 4.50.Dv, Lx I. INTRODUCTION Photons are well suited to be quantum information carriers [1]. Over the past decades, there has been a large number of both theoretically proposed and experimentally tested quantum information protocols designed for photons [ 4]. A notable example with practical applications is quantum cryptography, which allows for unconditionally secure transmission of information [5 9]. One can use both both fiber [10] and free-space [11] optics to distribute photon-encoded information over considerable distances. Even though photons are not so susceptible to interaction with the environment as, for instance, atoms [1], their state also deteriorates because of noise and absorption in the communication channel [13 15]. Since channel transmissivity and level of noise are limited by unavoidable technological imperfections, a viable alternative strategy to increase communication range is based on amplification. However, quantum properties of photon states (unless the state is known apriori) are not preserved by classical amplification based on a mere measure and resend or stimulated emission approach; thus these approaches are not always suitable [16]. Quantum amplifiers have to be used instead [17 ]. In discrete variable encoding, the polarization or spatial degree of freedom of individual photons is usually used to encode qubits. It is therefore not surprising that optical qubit amplifiers are proposed and built to address these degrees of freedom [3 8]. Similar to other linear-optical quantum gates [9], the qubit amplifiers are also probabilistic, and their successful operation has to be heralded by a specific detection outcome on ancillary photons. Thus apart from amplification gain, one has to introduce success probability to characterize the performance of qubit amplifiers. In general, a qubit amplifier performs the following transformation on a mixture of vacuum and single-qubit states: p p 1 ˆρ Q p 0 N 0 0 +p 1G N ˆρ Q, (1) * bartkiewicz@jointlab.upol.cz k.lemr@upol.cz FIG. 1. (Color online) Conceptual scheme of a heralding qubit amplifier. The input state is transformed according to Eq. (1). D, detector; EPR, ancillary photons; G, amplifier: FF, feed forward. where ˆρ Q and ˆρ Q stand for the input and output qubit density matrices, N denotes normalization, and G is the overall (nominal) gain of the amplifier (see conceptual scheme in Fig. 1). So far, only perfect amplifiers ( ˆρ Q = ˆρ Q ) have been discussed in the literature. In this paper, we extend the analysis of our previously published scheme [8] to the general case of imperfect amplification ( ˆρ Q ˆρ Q ). This paper is organized as follows: In Sec. II, we describe the principle of operation of the proposed scheme. Moreover we introduce the basic quantities used to characterize our proposed amplifier. We introduce the fidelity of the operation as the overlap between the input and output qubit states. This analysis allows us to establish the success probability versus fidelity trade-off and observe increased success probability at the expense of a fidelity drop that we describe in Sec. III. Finally, in Sec. IV, inspired by optimal state-dependent quantum cloning [30 3], we also show that having some aprioriinformation about the input state allows us to optimize the amplification procedure in order to improve this fidelity versus success probability trade-off. We conclude in Sec. V. II. PRINCIPLE OF OPERATION In this section we describe the principle of operation of our scheme depicted in Fig. so that in subsequent sections we can analyze the above-mentioned fidelity vs success probability trade-off and state-dependent amplification /013/88(6)/06304(7) American Physical Society S-61

136 BARTKIEWICZ, ČERNOCH, AND LEMR PHYSICAL REVIEW A 88, (013) FIG.. (Color online) Scheme for a state-dependent linearoptical qubit amplifier as described in the text. EPR, source of entangled ancillary photon pairs; PBS, polarizing beam splitter; PPBS, partially polarizing beam splitter (defined in the text); WP, wave plate; PDF, polarization-dependent filter; D, standard polarization analysis detection block (for reference see [33]). The signal state ψ s is prepared in superposition of vacuum 0 and polarization-encoded single-qubit state Q, ψ s =α 0 +β Q, () where ( α + β = 1) and the qubit Q =cos θ H +sin θ eiϕ V (3) is parametrized by angles θ and ϕ describing the superposition of horizontal H and vertical V polarization basis states. The amplifier also makes use of an ancillary pair of entangled photons in a state parametrized by angle χ [0; π 4 ], ψ a =cos χ HH +sin χ VV. (4) In the first step, the signal impinges on the first fully polarizing beam splitter PBSin, where the horizontal and vertical components of the signal qubit are separated into their respective modes. In these modes the interaction with the ancillary pairs of photons takes place: the horizontal component of the signal interacts with the first ancillary photon on a partially polarizing beam splitter PPBS1; similarly, the vertical signal component is combined with the second ancillary photon on the partially polarizing beam splitter PPBS. The partially polarizing beam splitter PPBS1 fully reflects vertically polarized photons and has reflectivity r for horizontal polarization. On the other hand, PPBS reflects all horizontally polarized light and has reflectivity r for vertical polarization. Partially polarizing beam splitter PPBS1 can be described in terms of creation operators â in,h râ out,h + 1 r â D1,H, â a1,h râ D1,H + 1 r â out,h, â a1,v â D1,V, where labeling of modes corresponds to the scheme in Fig.. Analogous transformation describes the action of the PPBS. Projection on diagonal D =( H + V )/ and antidiagonal A =( H V )/ linear polarization is performed in both detection modes D1 and D. The resulting signal state is recovered by combing horizontal and vertical components on the output fully polarizing beam splitter PBSout. One can trace how the individual components of the three-photon total state (signal and ancillary photons) get transformed by the setup assuming postselection on detection of one photon in each detection mode D1 and D: 0 in H a1 H a r 0 out H D1 H D, 0 in V a1 V a r 0 out V D1 V D, H in H a1 H a (r 1) H out H D1 H D, H in V a1 V a r H out V D1 V D, V in H a1 H a r V out H D1 H D, V in V a1 V a (r 1) V out V D1 V D. After the photons in the detection modes get projected to diagonal DD or antidiagonal AA linear polarization states (both detected photons share the same polarization), the output signal state can be expressed as ψ out1 = αr (cos χ + sin χ) 0 + βx + cos θ H +βy + sin θ eiϕ V, (5) where x ± = (r 1) cos χ ± r sin χ, (6) y ± = (r 1) sin χ ± r cos χ. The output state ψ out1 is intentionally kept unnormalized to provide a simple expression for success probability in subsequent calculations. Alternatively, the output signal state (also not normalized) takes the form of ψ out = αr (cos χ sin χ) 0 + βx cos θ H βy sin θ eiϕ V (7) if DA or AD coincidence is observed (detected photons have mutually orthogonal polarizations). A feed-forward operation has to be adopted to correct the qubit part of the state given by Eq. (5) to be identical to the qubit part of Eq. (7). This feed-forward transformation consists of polarization-dependent filtrations τ H and τ V when DD or AA coincidence is detected. These filtrations are functions of the ancilla parameter χ and reflectivity r but are signal state independent: τ H = x x +, τ V = y y +. (8) In the case of DA or AD coincidence detection, additional phase shift (sign flip) is imposed on vertical polarization (V V ). This process is not lossy, so we assume it is performed in all the subsequently evaluated scenarios. A. Success probability For the subsequent analysis, several quantities are crucial. The first is the overall success probability of the procedure S-6

137 STATE-DEPENDENT LINEAR-OPTICAL QUBIT AMPLIFIER PHYSICAL REVIEW A 88, (013) P succ. It can be expressed using the norm of the output state ψ out1 and ψ out. Not implementing the lossy feed-forward, the success probability reads P succ = ( ψ out1 ψ out1 + ψ out ψ out ) = α r + β x + + x cos θ + β y + + y sin θ, (9) where the factor of describes the two equally probable coincidences leading to ψ out1 or ψ out. On the other hand, if the feed-forward is implemented, the output states ψ out1 and ψ out are transformed to the forms ψ out1ff = αr (cos χ + sin χ) 0 + βx cos θ H +βy sin θ V, ψ outff = αr (cos χ sin χ) 0 + βx cos θ H +βy sin θ V, (10) and the corresponding success probability reads P succ = ( ψ out1ff ψ out1ff + ψ outff ψ outff ) ( = α r + β x θ cos + θ ) y sin. (11) B. Amplification gain A second very important parameter of the amplifier is the gain G, the ratio between the qubit and vacuum components for the amplified state divided by the analogous ratio for the initial input state, as shown in Eq. (1). In general, the gain can differ for horizontal and vertical polarizations. One can easily define the gain for both polarizations in the case where the feed-forward is implemented: G HFF = x r, G VFF = y r. (1) If the lossy feed-forward is not implemented, the gain can be calculated as the average gain for output states ψ out1 and ψ out, G H = x + + x, G r V = y + + y. (13) r The overall gain defined in Eq. (1) is obtained by combining the two gains for horizontal and vertical polarization. In the case of applied feed-forward, the overall gain is given by G FF = cos θ G HFF + sin θ G VFF, and in the case without the lossy feed-forward (only the phase flip is performed) it is given similarly by G = cos θ G H + sin θ G V. C. Amplification fidelity The last quantity that has to be calculated in this section is the output qubit fidelity F Q. This fidelity compares the overlap between the qubit state Q at the input with the qubit subspace of the output state ψ outq. If the feed-forward is implemented, the fidelity is simply ( x F QFF = ψ outq Q cos θ = + y ) sin θ x cos θ +. (14) y sin θ If only the feed-forward phase correction and not the full lossy transformation is performed, the fidelity of the output qubit reads F Q = Q ˆρ outq Q ( x+ cos θ = + y ) + sin θ ( + x cos θ + y ) sin θ (x+ + x ) cos θ +, (y + + y )sin θ (15) where ˆρ outq is the normalized density matrix of the singlephoton subspace, which is a balanced mixture of ψ out1 ψ out1 and ψ out ψ out, with the V V transformation performed on the latter. III. SUCCESS-PROBABILITY-FIDELITY TRADE-OFF In this section we investigate the trade-off between success probability P succ and the output-state fidelity F QFF. For this analysis, we fixed the parameters α = β = 1, and we also took into account the lossy feed-forward. A. Infinite gain First, we studied this trade-off on the particular case of infinite gain. The infinite gain is an important setting of qubit amplifiers. To achieve this regime, one simply sets r = 0. Thus the previously obtained expressions can be considerably simplified. Coefficients x + = x = cos χ and y + = y = sin χ become equal, so there is no need for lossy feed-forward any more (τ H = τ V = 1); only V V is performed. Success probability and qubit fidelity take the forms ( P succ = β cos χ cos θ + sin χ sin θ ) = β ( [cos χ θ ) ( + cos χ + θ )] (16) and ( cos χ cos θ F QFF = + sin χ sin θ ) cos χ cos θ +, (17) sin χ sin θ respectively. Figure 3 shows the dependence of the success probability on output-state fidelity for four different input states parametrized by θ ={π/,π/5,π/3,π/4} and ϕ = 0. The calculation reveals that there is no improvement in success probability in the case of a balanced input state (θ = π/), and the success probability remains constant and fidelity independent. In contrast to that, the more the input state is unbalanced, the S-63

138 BARTKIEWICZ, C ERNOCH, AND LEMR PHYSICAL REVIEW A 88, (013) confirming the finding described in Fig. 3. In addition, one can observe that when set to lower values of gain, the setup performs better for higher fidelities than for lower ones. In the case of higher gains, however, the setup behaves as described in the infinite-gain analysis. Also we were able to establish a state-dependent unreachable area, a set of gain and fidelity coordinates that cannot be reached by the presented setup. This area is visualized by the threshold (TRH) line shown in Fig. 4. IV. STATE-DEPENDENT AMPLIFICATION FIG. 3. (Color online) Success probability Psucc given by Eq. (16) as a function of output-state fidelity FQFF given by Eq. (17) in the case of infinite gain is depicted for four different input states as described in the text. more pronounced the dependence of the success probability on fidelity is. This fact will reemerge in Sec. IV, which discusses state-dependent amplification. For instance, in the case of θ = π/4, the success probability can be increased by a factor of 1.7 at the expense of 85% output-state fidelity. B. Maximum success probability In the next step, we performed a numerical calculation of the maximum achievable success probability for given values of overall gain given by Eq. (1) and the output-state fidelity given by Eq. (14). This calculation has been carried out on the same four input states mentioned above by varying the χ and r parameters. Plots in Fig. 4 present the obtained results, FIG. 4. (Color online) Success probability Psucc as a function of both output-state fidelity and amplification gain GFF is depicted for four different input states: (a) θ = π/, (b) θ = π/5, (c) θ = π/3, and (d) θ = π/4. THR stands for the threshold of unreachable area. This section brings forward the main result of the paper: how can we improve the success probability of amplification given some a priori knowledge about the input qubit state? For the purpose of quantifying the a priori information about the input signal, we use the von Mises Fisher distribution [34] (also known as the Kent distribution) describing dispersion on a sphere. This probability density function is defined as κ exp(κ cos θ ), (18) g(θ,κ) = 4π sinh(κ) where θ is the input-state parameter describing the axial angle of the state on the Poincare sphere and κ, i.e., the concentration parameter, determines the amount of knowledge about the input qubit. Figure 5 depicts the probability distribution over the Poincare sphere for various values of κ. Note that in the case of κ = 0, all states are equally probable (therefore no a priori knowledge), and the larger the concentration parameter κ is, the more precise the information about the input state is. This trend is illustrated in Table I, which provides the values of medians θm and first deciles θd for various values of κ. Note that while throughout this paper we center the FIG. 5. (Color online) Probability density function g = g(θ,κ) given by Eq. (18) over the Poincare sphere for various values of the κ parameter used in subsequent numerical simulations: (a) κ = 0, (b) κ = 1, (c) κ = 3, (d) κ = 10. Labels H, D, and R = ( H + i V )/ denote the position of the horizontal, diagonal, and right-hand circular-polarization states, respectively S-64

139 STATE-DEPENDENT LINEAR-OPTICAL QUBIT AMPLIFIER PHYSICAL REVIEW A 88, (013) TABLE I. Values of medians and first deciles of the von Mises Fisher distribution for several values of the concentration parameter κ. Parameter κ Median θ m (rad) First decile θ d (rad) 0 π/ 0.05π π 0.136π 3 0.0π 0.085π π 0.046π distribution g(κ,θ) around the northern pole of the sphere (horizontal polarization), the generality of our scheme does not suffer by this choice. If the knowledge about the input state is not centered around the north pole, one can always perform a deterministic rotation to make it so and inverse it after the state comes out of the amplifier. Using this quantification of input-state knowledge, we performed a series of numerical calculations with the goal being to determine the fidelity-success-probability trade-offs. Our results show the relation between the highest achievable average success probability, P succ = g(θ,κ)p succ dω, for the fixed values of average gain and fidelity, G = g(θ,κ)g FF dω, F = g(θ,κ)f QFF dω, respectively, where dω = dcos θdφ and is the surface of the Poincaré sphere. Only the F integral is not trivial since F QFF is a rational function of cos θ; thus it was calculated numerically. However, the other integrals can be expressed as linear functions of cos θ =coth κ 1/κ. The investigated cases are depicted in Fig. 6. In each case we targeted one specific average overall gain value from the set G {3dB,10 db,0 db, }, where the average was taken over input states distributed according to the von Mises Fisher distribution for four different values of κ {0; 1; 3; 10}. For all the average gain and κ combinations, we determined the relation between the average output-state fidelity and the average success probability. Note that similar to the previous section, we assumed α = β = 1 and we also took into account the lossy feed-forward. Similar to the case analyzed in Sec. III, not all the values of fidelity are accessible simply because the setup cannot produce fidelity lower than a certain threshold that depends on the values of κ and average gain. It is a very expected result that for the combination of κ = 0 and infinite gain, the success probability of the setup and fidelity are state independent. This result can be analytically verified using formulas from Sec. II for r = 0. In contrast, for other than infinite average gains, there is always a maximum of success probability depending on κ. Forκ, this maximum is found for unit fidelity F =1. It follows from the above-mentioned observations that for a given value of average gain G and κ, there exists a specific fidelity value F giving maximum success probability max F P =P max. In some cases this maximum FIG. 6. (Color online) Maximum achievable success probability P as a function of average fidelity F for various values of average overall gain [(a) G =3dB,(b) G =10 db, (c) G =0 db, (d) G ] and state knowledge described by parameter κ of the probability density function g = g(θ,κ)givenbyeq.(18). is to be found on the threshold providing the lower bound on the accessible fidelity values, but surprisingly, this is not always the case. This effect reflects the fact that the space of χ and r values providing, at the same time, the required value of the fidelity and the average gain has a nontrivial structure. Thus it seems that the question about the limits on the success rate of the state-dependent quantum amplifier for fixed amplification parameters does not have a simple answer. Nevertheless, it is apparent that, in general, one can increase the success probability of the setup at the expense of the lower success probability, but sometimes the maximum value P max can be reached at a lower cost than approaching the fidelity threshold. A. Merit function One can argue that some applications require perfect amplification with unit fidelity and thus it is not suitable to increase the success probability of the setup at the expense of lower fidelity. While this may indeed be true in some cases, realistic protocols for quantum communication have to be robust against at least some degree of fidelity drop. This leads us to formulate a figure of merit function inspired by [35] M = max{p succf } P succ (F = 1), (19) where the numerator is the maximum of the product of fidelity and the corresponding success probability and the denominator is just the success probability at unit fidelity. Since the product of fidelity and success probability can be understood as some sort of output rate of signal qubits, the function M gives the S-65

140 BARTKIEWICZ, ČERNOCH, AND LEMR PHYSICAL REVIEW A 88, (013) M M M M FIG. 7. (Color online) Merit function M given by Eq. (19) depicted for various parameters κ and average gains: (a) G = 3dB,(b) G =10 db, (c) G =0 db, (d) G. maximum factor of increased output signal rate if one allows for the fidelity to be smaller than 1 (see Fig. 7). It can be easily shown that for the very specific case of both infinite average gain and infinite κ, the setup gives exactly the same outcomes of the simple photon amplifier [36] based on the detect and reproduce method. However, for no apriori knowledge about the input state κ = 0, the setup provides the same functionality as a previously published qubit amplifier [8]. In this sense, the setup covers the transition between these two conceptually different devices. V. CONCLUSIONS The possibility to operate a qubit amplifier in an imperfect regime, where output qubit fidelity may be smaller than 1, offers a significant increase in success probability if one has some aprioriinformation about the input qubit state. In this paper, we analyzed the capabilities of the proposed linear-optical setup for the state-dependent qubit amplifier. We determined the output-state fidelity, gain, and success probability as functions of the setup parameters. Next, we performed a numerical optimization of success probability depending on target output-state fidelity and gain for various input states. This calculation shows that the closer the state is to the pole of a Poincaré sphere, the more pronounced the success probability improvement is if fidelity is allowed to drop. Also this effect manifests more strongly in the cases of higher gains. Furthermore we performed numerical analysis of success probability as a function of average output-state fidelity for several target average gains and levels of aprioriinformation about the input state quantified by the von Mises Fisher distribution [34]. The results show how the maximum success probability versus fidelity trade-off behaves depending on average gain and aprioriinformation about the input state. To clearly visualize the potential improvement in success probability, we have constructed a specific function of merit that we use to characterize the amplifier in several regimes (various gains and levels of aprioriknowledge about the input state). This analysis indicate that success probability can be increased on the order of tens of percent depending on the conditions. Interestingly, we found that, in general (for cases other than infinite gain), the success probability of the amplifier does not increase in a monotonic way for decreasing fidelity. This result clearly demonstrates that the success probability of statedependent amplifiers can be maximally increased without a significant drop in output-state fidelity. For this reason we believe that our results can stimulate further research on statedependent qubit amplifiers and their potential applications. If we draw a comparison between state-dependent amplification and cloning, we will notice a number of similarities. In this paper, we optimize the success probability of the former operation for a given target fidelity, gain, and apriori information. In contrast to amplification, cloning schemes are usually optimized to maximize fidelity given certain a priori information, disregarding other parameters such as the success probability of the cloning operation. In case of quantum cloning, the analysis of the fidelity vs success probability trade-off has been investigated, for instance, in a recent paper on cloning-based amplification [37]. In the relevant paper, fidelity and probability of success are optimized by switching between optimal probabilistic quantum cloning and deterministic classical cloning based on the measure and reproduce method. Drawing a direct comparison between the setup presented in this paper and the cloning-based amplifier is an intricate problem since they perform qualitatively different operations. The scheme presented in this paper removes the vacuum term from the qubit-vacuum superposition, while the cloning-based amplifier duplicates qubits, making the signal more resistant to attenuation. The presented analysis assumes the existence of perfect photon-number-resolving detectors and a perfect source of ancillary photons. In an experimental setting, however, one has to take into account the actual properties of laboratory setup components. Experimental imperfections will negatively affect the performance of the setup. For example, if the detectors do not resolve photon numbers, the amplifier will impose a lower gain than predicted. This is because in some cases it is possible for all photons to leak into detection modes, leaving a vacuum at the output; at the same time the operation is misleadingly heralded as successful. Such additional heralding would increase the success probability of the amplification. On the other hand, lower detection efficiency will not affect amplification gain, only lower the success rate. One of the key ingredients in our scheme is a pair of ancillary photons. Their preparation is crucial for this device to function correctly. The fidelity of the amplifier operation depends on how accurately this ancillary photon pair is prepared. Higher photon number contributions generated in the process of spontaneous parametric down-conversion can negatively affect amplification. In such cases the device may yield a false success and, instead of amplifying the input S-66