Bell tests in physical systems

Transcription

1 Bell tests in physical systems Seung-Woo Lee St. Hugh s College, Oxford A thesis submitted to the Mathematical and Physical Sciences Division for the degree of Doctor of Philosophy in the University of Oxford Trinity Term, 2009 Atomic and Laser Physics, University of Oxford

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3 Bell tests in physical systems Seung-Woo Lee, St. Hugh s College, Oxford Trinity Term, 2009 Abstract Quantum non-locality and entanglement in realistic physical systems have been of great interest due to their importance, both for gaining a better understanding of quantum physical principles and for applications in quantum information processing. Both quantum non-locality and entanglement can be effectively detected by testing Bell inequalities. Thus, finding Bell inequalities applicable to realistic physical systems has been an important issue in recent years. However, there have been several conceptual difficulties in the generalisation of Bell inequalities from bipartite 2-dimensional cases to more complex cases, which give rise to many fundamental questions about the nature of quantum non-locality and entanglement. In this thesis, we contribute to answering several fundamental questions by formulating new types of Bell inequalities and also by proposing a practical entanglement detection scheme that is applicable to any physical system. To start with, we formulate a generalised structure of Bell inequalities for bipartite arbitrary dimensional systems. The generalised structure can be represented either by correlation functions or by joint probabilities. We show that all previously known Bell inequalities can be written in the form of the generalised structure. Moreover, the generalised structure allows us to construct new Bell inequalities in a convenient way. Subsequently, based on this generalised structure, we derive a Bell inequality that fulfills two desirable properties for the study of high-dimensional quantum non-locality. The first property is the maximal violation of Bell inequalities by maximal entanglement which agrees with the intuition of maximal violation of local i

4 ii realism by maximal entanglement. The second property is that the Bell inequality written in correlation space should exactly represent a boundary between quantum mechanics and local realism. In contrast to any previously known Bell inequality, the derived Bell inequality is shown to satisfy both conditions. We apply this Bell inequality to continuous variable systems and demonstrate maximal violation by the maximally entangled state associated with position and momentum. We then formulate a generalised Bell inequality in terms of arbitrary quasiprobability functions in phase space formalism. This includes previous types of Bell inequalities formulated using the Q and Wigner functions as limiting cases. We show that the non-locality of a quantum system is not directly related to the negativity of its quasi-probability distribution beyond the previously known fact for the case of the Wigner function. We also show that the Bell inequality formulated using the Q-function permits the lowest detector efficiencies out of the quasi-probability distributions considered. Finally, we present a general approach for witnessing entanglement in phase space by significantly inefficient detectors. Its implementation does not require any additional process for correcting errors in contrast to previous proposals. Moreover, it allows detection of entanglement without full a priori knowledge of detection efficiency. We show that entanglement in single photon entangled and two-mode squeezed vacuum states is detectable by means of tomography with detector efficiency as low as 40%. This approach enhances the possibility of witnessing entanglement in various physical systems using current detection technologies.

5 Contents Abstract i Chapter 1. Introduction 1 Chapter 2. Basic Concepts High-dimensional quantum systems Quantum states in high-dimensional Hilbert space High-dimensional physical systems Bell s theorem Bell s theorem and Bell inequalities Bell test experiments Loopholes in Bell tests Polytope representation of Bell s theorem Quantum entanglement Entanglement Genuine entanglement Entanglement Witness Phase space representations Generalised quasi-probability functions Bell inequalities in phase space iii

6 iv Contents Chapter 3. Generalised structure of Bell inequalities for arbitrarydimensional systems Introduction Generalised arbitrary dimensional Bell inequality Violation by Quantum Mechanics Tightness of Bell inequalities Remarks Chapter 4. Maximal violation of tight Bell inequalities for maximal entanglement Introduction Optimal Bell inequalities Extension to continuous variable systems Conclusions Chapter 5. Testing quantum non-locality by generalised quasi-probability functions Introduction Generalised Bell inequalities of quasi-probability functions Testing Quantum non-locality Violation by single photon entangled states Violation by two-mode squeezed states Discussion and Conclusions Chapter 6. Witnessing entanglement in phase space using inefficient detectors Introduction Observable associated with efficiency

7 Contents v 6.3 Entanglement witness in phase space Testing single photon entangled states Testing two-mode squeezed states Testing with a priori estimated efficiency Conclusions Chapter 7. Conclusion 95 Bibliography 99

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9 Chapter 1 Introduction Bell s theorem [1] has been called the most profound discovery of science [2] and also one of the greatest discoveries of modern science [3]. These words highlight the importance of the quantum features i.e. quantum non-locality and entanglement that were then found based on Bell s theorem. Quantum non-locality and entanglement show striking properties that can not be understood in the context of classical physics, and thus shift the paradigm of understanding fundamental principles in physical systems. Moreover, these quantum features are a promising resource for quantum information processing that is a revolutionary technology superior to classical counterpart in various ways [4]. In these senses observing quantum non-locality and entanglement in physical systems can be seen as a surprising discovery. Historically, the concepts of quantum non-locality and entanglement appeared for the first time in the famous Einstein-Podolski-Rosen (EPR) paradox [5] 1. This paradox was formulated with the theory of local realism in mind to prove the incompleteness of quantum mechanics. Later Bell showed in his theorem that local realism leads to constraints on correlations between measurement results carried out on two separated systems [1]. These constraints, known as Bell inequalities, provide a possible method for testing the validity of quantum mechanics against local realism. It 1 The concept of entanglement was also introduced by E. Schrödinger [6, 7]. 1

10 2 Introduction was theoretically shown that Bell inequalities are violated by the quantitative predictions of quantum mechanics in the case of entangled states [1, 8] 2. Subsequently, experiments have confirmed the validity of quantum mechanics by demonstrating the violation of Bell inequalities [9, 10]. In general, quantum non-locality and entanglement manifest themselves in such a counterintuitive way, precisely through the violation of the local realism [9, 10] and by permitting stronger correlations beyond the level allowed in classical physics [11]. Quantum non-locality and entanglement in simple models such as bipartite or tripartite 2-dimensional systems are currently well understood [11]. However, little is known about them in realistic physical systems composed of many particles with many degrees of freedom, which we will call complex systems throughout this thesis. In fact most physical systems in nature are complex systems, and thus generalisations of Bell s theorem to complex systems have been regarded as one of the most important challenges in quantum mechanics [12, 13, 14, 15, 16, 17, 18, 19, 20, 11, 21, 22]. There are several motivations for studying Bell s theorem in complex systems. Firstly, generalising Bell s theorem to complex systems would provide a way to observe quantum features in the macroscopic world. Macroscopic systems are complex systems that are generally governed by classical physics, and thus observing quantum properties in those systems seems to be difficult. However, recent studies on e.g. the relation of entanglement with macroscopic observables [21] offer possibilities to investigate quantum properties in macroscopic systems [23]. For example, Bell type inequalities constructed with macroscopic observables would allow one to detect quantum non-locality and entanglement in macroscopic physical systems. Secondly, Bell inequalities can help us to investigate quantum phenomena arising in realistic physical systems. For example, studying the role of entanglement in a 2 Note that not all entangled states violate Bell inequalities.

11 3 quantum phase transition is one of the most interesting issues in recent relevant research [24]. In fact, quantum phase transition can be regarded as a change of the dominant degrees of freedom of a system. Thus Bell inequalities defined in both degrees of freedom that the system traverse would be an effective tool for the study of entanglement in the vicinity of a quantum phase transition. This can be also used for studying, for example, entanglement transfer between different physical degrees of freedom. Thirdly, studying quantum non-locality and entanglement in realistic physical systems is essential for applications in quantum information processing. Quantum non-local properties and entanglement have been considered as essential resources for various quantum information processing protocols [25, 11, 20]. All candidates of quantum information processing are complex systems [20, 21, 26, 27]. Although a specific degree of freedom is chosen as the basic unit of quantum information processing, i.e. a qubit, the other degrees of freedom can still affect the qubit and should be considered in realistic implementations. Thus investigation of a qubit system in high-dimensional formalism might be useful to understand the behaviour of qubits in realistic implementations. Furthermore, implementations of quantum information processing in complex systems can provide practical advantages. For example, high-dimensional quantum cryptography can be more secure than 2-dimensional cases [28, 29, 30]. Quantum teleportation, quantum computation, and quantum cryptography can be efficiently implemented by optical continuous variable systems [20, 31]. In spite of these motivations, realisation of Bell tests in complex physical systems still suffers from both conceptual and technical difficulties, and there are numerous relevant open questions [16, 15]. Yet we do not have a full picture of all possible manifestations of quantum phenomena in complex systems. For example, there have been no known Bell inequalities for high-dimensional systems that are maximally vi-

12 4 Introduction olated by maximally entangled states without bias at the degree of violation [32, 33]. We still do not have a clear picture of quantum non-locality in phase space formalism and its relations to other quantum properties. Moreover, there is no general method for quantifying entanglement in complex systems. Due to several effects arising in complex systems, it is difficult to observe quantum behaviour clearly. For example, since complex physical systems are generally macroscopic and interact strongly with their environment, decoherence can influence entanglement properties in those systems. Moreover, with increasing size or dimensionality, a precise measurement of a system becomes difficult, especially if one wants to preserve its quantum properties. This is often due to detector imperfections. In this thesis we focus on several topics related to Bell tests in realistic physical systems, which shall be described as follows. The first topic studied in this thesis is the extension of Bell inequalities to arbitrary dimensional bipartite systems. We consider two desirable conditions of Bell inequalities in order to investigate quantum non-locality properly in high-dimensional quantum systems. For 2-dimensional systems the Clauser-Horne-Shimony-Holt (CHSH) inequality [8] has the desirable property of only being maximally violated for a maximally entangled state. In addition, the CHSH inequality is a tight Bell inequality i.e. a facet of the polytope defining the boundary between local realism and quantum mechanics. This means that any violation of local realism on this particular facet is indicated by the CHSH inequality [33]. Note that tightness is a desirable property since only sets of tight Bell inequalities can provide necessary and sufficient conditions for the detection of pure state entanglement. For d-dimensional systems with d > 2 there has been no known Bell inequality to satisfy both desirable properties so far. For example, the Bell inequality proposed by Collins et al. [13] is maximally violated by nonmaximal entanglement [32] and the Bell inequality in the case of Son et al. [14] was shown to be non-tight [17]. Therefore, it would be ideal to find a Bell inequality

13 5 satisfying both conditions for arbitrary d-dimensional systems. In addition, such a Bell inequality may provide practical advantages in e.g. the preparation of an ideal channel for higher-dimensional quantum teleportation [34] or cryptography [25]. We will address this topic in chapter 3 and 4, where we formulate a generalised structure of Bell inequalities for bipartite arbitrary d-dimensional systems and derive a Bell inequality that fulfills two desirable properties: maximal violation by maximally entangled states and tightness. Another topic of this thesis is to study quantum non-locality in phase space formalism. Phase space representations are a convenient tool for investigating quantum states as they provide insights into the boundaries between quantum and classical physics. Any quantum state ˆρ can be fully characterised by the quasi-probability function [35, 36]. The negativity of the quasi-probability function has been regarded as a non-classical feature of quantum states, and is thus believed to have a fundamental relation to quantum non-locality. Bell argued [37] that the original EPR state [5] will not exhibit non-locality since its Wigner-function is positive everywhere and hence serves as a classical probability distribution for hidden variables. However, later Banaszek and Wódkiewicz showed how to demonstrate quantum non-locality using the Q- and Wigner-functions [38, 19, 39]. Remarkably, this showed that there is no direct relation between the negativity of the Wigner function and quantum non-locality. Since then, in spite of various efforts to explain more precisely the relation between quantum non-locality and the negativity of quasi-probability functions [40], a clear answer has been missing. This is mainly because we still do not have a general method to describe quantum non-locality in phase space formalism. In particular, a Bell inequality formulated by generalised quasi-probability functions would be necessary, by which one could demonstrate how non-locality changes the extent of the negativity. We will propose such a method in chapter 5 to test Bell inequalities using arbitrary quasi-probability

14 6 Introduction functions and study the relation between quantum non-locality and the negativity of quasi-probability functions. Our final topic is to find an efficient detection scheme for entanglement in phase space formalism. Entanglement detection is one of the primary tasks both for investigating fundamental aspects of quantum systems and for applications in quantum information processing [11]. However, in most cases, experimental realisations of testing entanglement suffer from detector imperfections since measurement errors wash out quantum correlations. This difficulty becomes more significant as the size or dimensionality of the systems increases. This is unfortunate as entanglement in larger systems is gaining more attention [20, 21]. For example, the violation of a Bell type inequality for continuous variable systems e.g. two-mode squeezed states [19, 41] requires almost perfect photo-detection efficiency [42]. Several schemes have been proposed to overcome this problem such as e.g. numerical inversion of measured data [43] and iterative reconstruction methods [18, 44], but require a large number of calculation or iteration steps. Therefore, an entanglement detection scheme that is practically usable in the presence of noise is necessary. Furthermore, a general entanglement criterion is required, which is applicable independently of the particular physical systems in phase space. Such an entanglement criterion would also be essential in entanglement based quantum information protocols [20]. For example, preparing an entangled channel is a necessary precondition for any secure quantum key distribution protocol [45]. In chapter 6 we will propose a detection scheme of entanglement using inefficient detectors. As motivated from these three topics, we investigate quantum non-locality and entanglement in physical systems. Our research aims to provide answers the questions arising in those topics. Detailed descriptions of each chapter are presented as follows.

15 7 In chapter 2, we will review basic concepts that are necessary for describing most of the content in this thesis. We start by introducing high-dimensional quantum systems. Then, the basic concepts of quantum non-locality will be explained in the context of the original Bell theorem. We consider loophole problems that occur in experimental Bell tests. We also introduce the polytope representation of Bell s theorem that is useful to extend Bell inequalities to complex systems. Then, we compare entanglement witnesses, which discriminate entangled states from separable states, to Bell inequalities. Finally, we introduce the phase space representation of quantum states and a Bell inequality represented in this formalism. In chapter 3 we will formulate a generalised structure of Bell inequalities for bipartite arbitrary d-dimensional systems, which can be represented either in terms of correlation functions or joint probabilities. The two known highdimensional Bell inequalities proposed by Collins et al. [13] and by Son et al. [14], will be considered in the framework of the generalised structure. We then investigate the properties of these Bell inequalities with respect to the degree of entanglement. In chapter 4 we will derive a Bell inequality for even d-dimensional bipartite quantum systems that fulfills two desirable properties: maximal violation by maximally entangled states and tightness. These properties are essential to investigate quantum non-locality properly in higher-dimensional systems. Then we apply this Bell inequality to continuous variable systems and investigate its violations. We also discuss which local measurements lead to maximal violations of Bell inequalities in continuous variable Hilbert space. In chapter 5 we will investigate quantum non-locality in phase space formalism. We first formulate a generalised Bell inequality in terms of the s-parameterised

16 8 Introduction quasi-probability function. We then demonstrate quantum non-locality by the single-photon entangled and two-mode squeezed states as varying the parameter s and detection efficiencies. We finally discuss the relation between quantum non-locality and the negativity of quasi-probability functions. In chapter 6 we propose an entanglement detection scheme using significantly inefficient detectors. This is applicable to arbitrary quantum states described in phase space formalism. We formulate an entanglement witness in the form of a Bell-like inequality using directly measured Wigner functions. For this, we include the effects of detector efficiency into possible measurement outcomes. Using this entanglement witness, we detect entanglement in the single photon entangled and two-mode squeezed states with varying detection efficiency. We finally discuss the effects of a priori knowledge of detection efficiency on the capability of our scheme. We conclude this thesis in chapter 7 and give an outlook on the directions of further research. Apart from the research presented in this thesis, I have also contributed to propose entanglement purification protocols which are applicable to multipartite high-dimensional systems [46].

17 Chapter 2 Basic Concepts In this chapter, we present basic ideas for the study of Bell s theorem in physical systems. We review a high-dimensional description of quantum states, and define high-dimensional physical systems. Basic concepts of Bell s theorem and its experimental implementations are presented. As considering implementations we introduce loophole problems arising in realistic Bell tests. We also introduce a polytope representation of Bell inequalities that is useful to study Bell s theorem in complex systems. For the extension of Bell s theorem to the phase space formalism we review Bell inequalities formulated by quasi-probability functions. In addition, we also consider entanglement witnesses as comparing to Bell inequalities. 2.1 High-dimensional quantum systems Quantum states in high-dimensional Hilbert space Pure quantum states of a system are represented by vectors in Hilbert space. A system completely described in d-dimensional Hilbert space, H d, is called a d- dimensional quantum system (or equivalently d-level quantum system). For the simplest example, a 2-dimensional quantum system is described in the 2-dimensional Hilbert space H 2 spanned by two orthonormal basis vectors, ψ 0 and ψ 1. An ar- 9

18 10 Basic Concepts bitrary superposition of two basis vectors ψ = α ψ 0 + β ψ 1 is also a possible state of the system, where their amplitudes α and β are arbitrary complex numbers. The normalisation condition for the state ψ i.e. α 2 + β 2 = 1, leads us to interpret α 2 and β 2 as the probabilities that the system is measured to be in states ψ 0 and ψ 1 respectively. This state can be regarded as a basic unit of quantum information processing i.e. a qubit. Then, the two computational basis vectors are labeled by ψ 0 = 0 and ψ 1 = 1. The computational basis can be arbitrarily chosen by transformation in 2-dimensional Hilbert space. For example, + = ( )/ 2 and = ( 0 1 )/ 2 constitute another orthonormal basis of qubits. In general, the unitary transformations in 2-dimensional Hilbert space, SU(2), provide an infinite number of possible basis sets. While a pure quantum state is described by a single vector as described above, a mixed state is given as a statistical ensemble of pure states ˆρ = i p i ψ i ψ i where p i is the probability that the system in the state ψ i. The concept of a mixed state comes up when the state of a system is not exactly known but given as a mixture of different states. The most general description of a 2-dimensional system is given as a density matrix ˆρ = 1(1 + a σ) where a = (a 2 1, a 2, a 3 ) is a real vector and σ = (ˆσ x, ˆσ y, ˆσ z ) is a vector of pauli operators. This can be visualised by the Bloch sphere which represents all possible states of a single qubit [4]. Pure quantum states correspond to the surface of the sphere, that is a 2 = 1, while mixed states correspond to the interior region of the sphere. A high-dimensional system is defined as a system which should be described in more than 2-dimensional Hilbert space. Its mathematical description is straightforwardly extended from the 2-dimensional formalism. We consider a d-dimensional (d > 2) Hilbert space spanned by d orthonormal basis vectors, { 0, 1,..., d 1 }. All pure states in d-dimensional Hilbert space can be represented using the basis

19 2.1. High-dimensional quantum systems 11 vectors d 1 ψ = α k k, (2.1) k=0 where the complex numbers α k satisfy the condition k α k 2 = 1, and α k 2 is the probability that the system is found to be in the state k. This can be regarded as the quantum version of a d-dimensional computational basic unit i.e. a qudit. Like in the 2-dimensional case, the qudit basis can be freely chosen by an arbitrary unitary transformation in d-dimensional Hilbert space, SU(d). In general an arbitrary quantum state in d-dimensional Hilbert space can be described as ˆρ = 1 d2 1 d (1 + i=0 a iˆλi ), (2.2) where a i = Trˆλ i ˆρ and a 0 = 1. Here λ = (ˆλ 0,..., ˆλ d 2 1) is a generalised pauli operator in d-dimensional Hilbert space and the vector a = (a 0,..., a d 2 1) is a d 2 -dimensional real vector. This is the so called generalised Bloch representation of d-dimensional quantum states [47]. The generalised Pauli operators, ˆλ i, in d-dimensional Hilbert space, H d, are given by [48, 49] ( ˆX d ) a (Ẑd) b, a, b 0, 1,...d 1, (2.3) where ˆX d and Ŷd are defined as d 1 ( 2πi ˆX d = k + 1 k, Ẑ d = exp d k=0 ) ˆN. (2.4) Here ˆN = k k k k is the number operator, and ˆX d and Ŷd transform the d-

20 12 Basic Concepts dimensional computational basis by ˆX d k = k + 1, ( ) 2πi Ẑ d k = exp d k k. (2.5) Note that ˆX d and Ẑd do not commute as Ẑd ˆX d = exp(2πi/d) ˆX d Ẑ d. The generalised Pauli operators in Eq. (2.3) provide a basis for arbitrary unitary operations in d- dimensional Hilbert space, which we will use to change the measurement basis in high-dimensional Bell tests High-dimensional physical systems General physical systems are composed of many particles with many degrees of freedom. In a given physical system one may be interested in a specific degree of freedom which we will call the target degree of freedom, or specific subsystems which we will call target subsystems. Based on these concepts we define the following cases as high-dimensional physical systems. First, most degrees of freedom in physical systems are represented by superposed states with more than 2 possible outcomes i.e. high-dimensional. For example, the position and momentum of a free particle are continuous variables in infinitedimensional Hilbert space. The angular momentum of an electron in atoms is finite high-dimensional. Therefore, if the target degree of freedom that we are interested in is high-dimensional, we regard the corresponding system as a high-dimensional system. Second, most physical systems have many degrees of freedom. In realistic implementations, it is very hard to single out a specific degree of freedom properly so that any removed degrees of freedom do not affect considerably the degree of freedom being singled out. One might consider the effects of other degrees of freedom as noises caused by complexity. However, in order to understand properly several

21 2.1. High-dimensional quantum systems 13 quantum phenomena in realistic systems, it is necessary to consider more than two degrees of freedom simultaneously and their influences on each other. For example, a quantum phase transition can be generally understood as an abrupt change of the dominant degree of freedom by varying an external parameter in physical systems [50] as shown in the case that cold atoms in optical lattices traverse two states, Mott-insulator and superfluid [51]. Therefore, if we consider multiple degrees of freedom simultaneously in a given system, their properties should be described in the framework of a high-dimensional formalism. Thus the corresponding system is high-dimensional. Third, most physical systems are composed of many particles. Thus it is also very hard to single out the target subsystems properly so that the removed rest subsystems do not affect them considerably. In certain cases the target subsystems can be selected as open systems and the effects from rest subsystems can be regarded as effects of the environment. In most cases, in order to investigate effectively many body systems, it is necessary to choose target subsystems as collective bodies of many particles which should be described in high-dimensional Hilbert spaces. For example, a subsystem composed of N qubits can be considered in the d = 2 N dimensional Hilbert space [52]. Therefore, we can consider the systems with a higher-dimensional target degree of freedom, with multiple target degrees of freedom, having target subsystems composed of many particles as high-dimensional systems. Several quantum features in such systems, which might not arise in e.g. 2-dimensional bipartite systems, can be effectively investigated in the framework of the high-dimensional formalism. For example, entanglement of

22 14 Basic Concepts multiple target degrees of freedom, which is called as the hyper entanglement, has been realised and investigated based on the high-dimensional formalism [53, 54]. Polarisation, time-bin and spatial degree of freedom have been used to create highdimensional systems with d = 3 [55], d = 4 [56], and d = 8 [57]. In addition, highdimensional systems are applicable to quantum information processing and provide some advantages e.g. a robust quantum key distribution [28, 29, 30], superdense coding [58], fast high fidelity quantum computation [59, 60, 61]. To summarise, physical systems existing in nature are high-dimensional systems, and several quantum phenomena in such systems can be effectively investigated in the framework of high-dimensional Hilbert space. This may lead to obtaining fundamental insight into the properties of complex quantum systems. Moreover, their properties are also applicable to quantum information processing. 2.2 Bell s theorem Bell s theorem and Bell inequalities Bell introduced a theorem about quantum non-locality in his seminal paper entitled On the Einstein-Podolsky-Rosen paradox [1]. In his theorem he discussed the famous paradox presented by Einstein, Podolsky, and Rosen (EPR) [5], which was intended to prove the incompleteness of quantum mechanics. Bell s argument begins with two assumptions as asserted in the EPR paper: Reality - the measurable quantity must have a definite value before the measurement takes place. Locality - the physical quantities within reality would not influence each other at a large distance.

23 2.2. Bell s theorem 15 Based on these assumptions together, called local realism, Bell derived a constraint in the form of an inequality which limits the correlational expectation values of measurement outcomes for two spatially separated parties. This is called the Bell inequality. Any violation of this inequality by the quantitative prediction of quantum mechanics implies that at least one of the two assumptions, reality or locality, must be abandoned. The most considerable achievement of Bell s argument is to provide a possible method for testing the validity of quantum mechanics against local realism. Thus, it was thought that Bell s theorem may put an end to the debate between local realism and quantum mechanics. However, it took more time to realise a test of Bell inequality due to several technical difficulties. With the progress of quantum control techniques, finally the violation of Bell inequalities has been demonstrated [9] as clear evidence of existing quantum non-local properties which defeats local realism. Ever since the EPR argument and Bell s theorem, many versions of Bell inequalities have been derived similar to Bell s original inequality. The most famous version was proposed by Clauser, Horne, Shimony and Holt (CHSH) [8]. Suppose that two spatially separated parties named Alice and Bob perform measurements independently. Each observable can be chosen from two possible settings denoted by A 1, A 2 for Alice and B 1, B 2 for Bob. There is no influence on the measurement selection between two parties. We here assume that all observables have two possible outcomes ±1, i.e. 2-dimensional outcomes. Then we can consider a combination of all possible correlations, A 1 B 1, A 1 B 2, A 2 B 1, and A 2 B 2, as A 1 B 1 + A 1 B 2 + A 2 B 1 A 2 B 2 = ±2, (2.6) which can take either the deterministic value +2 or 2 depending on measurement outcomes of each observable. Let us define the joint probability P (a 1, a 2, b 1, b 2 )

24 16 Basic Concepts which indicates that the system is in a state where A 1 = a 1, A 2 = a 2, B 1 = b 1, and B 2 = b 2 before the measurement. Then the average of the combination in Eq. (2.6) is written by E(A 1 B 1 + A 1 B 2 + A 2 B 1 A 2 B 2 ) = a 1,a 2,b 1,b 2 P (a 1, a 2, b 1, b 2 )(a 1 b 1 + a 1 b 2 + a 2 b 1 a 2 b 2 ) = a 1,a 2,b 1,b 2 P (a 1, a 2, b 1, b 2 )a 1 b 1 + P (a 1, a 2, b 1, b 2 )a 1 b 2 +P (a 1, a 2, b 1, b 2 )a 2 b 1 P (a 1, a 2, b 1, b 2 )a 2 b 2 = E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ). (2.7) From Eq. (2.6) and Eq. (2.7), we can obtain an inequality 2 E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ) 2, (2.8) which is called the CHSH inequality. Note that the upper and lower bound of the CHSH inequality are given as the deterministic maximum and minimum value of the Eq. (2.6). Let us now assume that two parties share a quantum state ψ = 1 2 ( ). (2.9) Each party can choose observables A 1 = S A a 1 and A 2 = S A a 2 for Alice, and B 1 = S B b 1 and B 2 = S B b 2 for Bob as varying the unit vectors of a 1, a 2, b 1, and b 2. Here S A and S B are the spin operator defined as S = (S x, S y, S z ) which is proportional to the pauli operators. If the chosen observables are S A a 1 = S z, S A a 2 = S x, S B b 1 = (S z + S x )/ 2, and S B b 1 = (S z S x )/ 2, the expectation

25 2.2. Bell s theorem 17 value for the quantum state in Eq. (2.9) is given by A 1 B 1 + A 1 B 2 + A 2 B 1 A 2 B 2 = 2 2, (2.10) which violates the CHSH inequality (2.8). This is a remarkable result which shows the existence of the non-local correlations in the state Eq. (2.9). It is also inevitably shown that local realism should be abandoned. 1 Let us then consider another version of Bell inequality proposed by Clauser and Horne (CH) in 1974 [65] as 1 P (X 1, Y 1 ) + P (X 1, Y 2 ) + P (X 2, Y 1 ) P (X 2, Y 2 ) P (X 1 ) P (Y 1 ) 0,(2.11) where P (X 1, Y 1 ) is the joint probability for the local measurement X 1 and Y 1, and likewise for others. It can be derived from the CHSH combination given in Eq. (2.7) 2 a 1 b 1 + a 1 b 2 + a 2 b 1 a 2 b 2 2, (2.12) with outcome variables defined by x 1 = (1 + a 1 )/2, x 2 = (1 + a 2 )/2, y 1 = (1 + b 1 )/2 and y 2 = (1 + b 2 )/2, x 1, x 2, y 1, y 2 {0, 1}. We can then obtain another form of inequality from Eq. (2.12) as 4 a 1 b 1 + a 1 b 2 + a 2 b 1 a 2 b 2 2 = 4(x 1 y 1 + x 1 y 2 + x 2 y 1 x 2 y 2 x 1 y 1 ) 0. (2.13) Finally, we find that the statistical average of this combination gives the CH inequality given in Eq. (2.11). 1 There are several other interpretations which explain this result e.g. the non-local hidden variable model by David Bohm [62, 63] and many world interpretations [64]. Details of these interpretations are beyond the scope of this thesis.

26 18 Basic Concepts Bell test experiments The test of the CHSH inequality can be implemented by various 2-dimensional systems with measurements of 2-dimensional outcomes. For example, one can consider the Stern-Gerlach measurement for entangled spin pairs or the polarisation measurement for entangled photon pairs. Let us here consider a Bell test performed by an optical setup to measure polarisations of entangled photon pairs [9]. Suppose that Alice and Bob are spatially separated and share entangled photon pairs which are generated from an optical source. Each photon goes through a polarising beam splitter whose orientation can be freely chosen by each party as shown in Fig Photons are detected at two output channels of the polarising beam splitter. We assume that measured data is recorded only for coincident detections at both parties. Possible outcomes for each are denoted by + or, and thus there are four possible compound data for a single trial: ++, +, +, and. After many trials of experiments with a specified measurement setting we can obtain the joint probability by statistical average as P (+ + ab) = N ++ N, (2.14) where N ++ is the number of detections with outcomes ++ and likewise for others, and N = N ++ + N + + N + + N. Here a and b denote the measurement setting changed by rotating the polarisation at the beam splitter. We then define the correlation function between measurement results of a and b as E a,b = P (+ + ab) P (+ ab) P ( + ab) + P ( ab). (2.15)

27 2.2. Bell s theorem 19 Figure 2.1. Optical setup for the Bell test (CHSH inequality). Entangled photons generated from the source are distributed between two separate parties. Photons go through a beam splitter and are measured at the detector with coincidence counting. After measurements for all a, b = 1, 2, we can obtain E 1,1, E 1,2, E 2,1, and E 2,2. Finally the statistical average of the CHSH combination can be obtained as E 1,1 + E 1,2 + E 2,1 E 2,2 2, (2.16) which should be bounded by the value 2 in the local realistic theories. Any statistical average exceeding this bound guarantees that the shared states have non-local properties. Let us consider the case when the shared photon pairs are in the state ψ = 1 2 ( ). (2.17) The measurement basis are varying by rotation from a fixed direction associated with the standard basis + and in a plane. Therefore, the measurement basis for Alice are written as θ a, + = cos θ a + sin θ a and θ a, = sin θ a + + cos θ a, and for Bob as φ b, + = cos φ b + sin φ b and φ b, + = sin φ b + + cos φ b. From Eq. (2.14) we can measure the joint probability that has the expectation value as P (+ + ab) = ψ θ a, + φ b, + 2 and likewise for others.

28 20 Basic Concepts The correlation function in Eq. (2.15) can be then obtained as E a,b = cos 2(θ a φ b ), a, b = 1, 2. (2.18) If the measurement basis are chosen as θ 1 = 0, θ 2 = π/4 for Alice and φ 1 = π/8 and φ 2 = π/8 for Bob, the expectation value of the CHSH combination can exceed the local realistic bound 2 (it reaches 2 2 in the case of perfect measurements). Therefore, one can observe the violation of the CHSH inequality experimentally. Since Bell s theorem, a great number of Bell test experiments have been performed and confirmed quantum mechanics against local realistic theories. The first test of Bell s theorem was performed in 1972 by Freedman and Clauser [66]. They demonstrated violations of a CH-type inequality by polarisation correlations of photons emitted by calcium atoms. Later, a test of the CHSH inequality was implemented by Aspect et al. in 1982 [9]. 2 Tittel et al. [69] and Weihs et al. [70] conducted Bell tests using photon pairs that were space-like separated. Progresses of quantum technologies have led to significant improvement in both efficiencies and variety of Bell tests. However, in realistic implementations of Bell tests, there exist several conceptual difficulties as we will explain in the following subsection Loopholes in Bell tests Beyond technical difficulties in the realisation of Bell tests, there exist also some conceptual difficulties. Indeed, so far there have been no experimental demonstrations of quantum non-locality without supplementary assumptions. Thus, there are still many scientists who point out the fact that the violation of Bell inequalities can be explained as faults of experimental setup in local realistic theories. This is the 2 Aspect et al. also conducted other versions of Bell type inequalities [67, 68] such as the type proposed by Clauser and Horne [65] and original Bell inequality [1].

29 2.2. Bell s theorem 21 so called loophole problem of the Bell test. Let us here consider two main loophole problems as follows: locality loophole - This arises from the difficulty in separating two local particles. When the distance between two local measurements is small it conflicts with the assumption of no communication between one observer s measurements to the other. Therefore, to prevent such a loophole problem the two parties should be space-like separated. However, it is also technically difficult to separate two sub-systems without losing their quantum properties. It seems that the most promising candidate for closing this loophole problem is optical photons [69] as other massive particles are difficult to separate sufficiently so that space-like measurements can be performed. [10]. In 1998, Weihs et al. conducted, for the first time, a Bell test experiment that closed the locality loophole using photons. In their scheme, the choice of local polarisation measurement was ensured to be random in order to avoid any connection between separated measurements [70]. detection loophole - All experiments suffer from the imperfection of realistic detectors. This causes the so called detection loophole problem. The inefficiency of detectors makes a violation of Bell inequalities compatible with local realism. Let us consider the Bell test scheme described in the previous section with detection efficiency less than 100%. In this scheme measured data is recorded when particles are detected coincidentally. However, a coincident measurement is affected by the detection efficiency. Thus the effect changes the CHSH inequality into [71] E1,1 coinc + E1,2 coinc + E2,1 coinc E2,2 coinc 4 2, (2.19) η

30 22 Basic Concepts where η is the detection efficiency. For the quantum states given in Eq. (2.9), the left-hand side still can reach the value 2 2. Therefore, the violation of Eq. (2.19) can be demonstrated only if η > 2( 2 1) This means that in order to demonstrate quantum non-locality without any supplementary assumption highly efficient detectors (η > 0.83) are required. Scientists usually assume that the sample of detected pairs is representative of the pairs generated at the sources, i.e. the fair sampling assumption that reduces the right-hand side in Eq. (2.19) to 2. However, it is a supplementary assumption in addition to local realism which prevents us observing genuine non-local properties. The minimum required efficiency for a non-locality test has been investigated for various measurement setups and systems [72, 73, 74, 75]. Numerous efforts have attempted to close the detection loophole problem, which may lead to the so called loophole-free Bell test [76, 77]. For example, Rowe et al. [10] performed Bell tests using massive ions instead of photons with 100% efficiency, though it has a locality loophole problem. These loophole problems have been also taken seriously in applications to quantum information processing. For example, the detection loophole affects the security of quantum key distribution protocols [25]. In the presence of detection loophole several attacks are possible in the protocols of quantum key distribution [78, 79]. The threshold of efficiency for secure protocols has been studied, for instance, it has been shown that the overall efficiency taking into account the channel loss and the detection efficiency together is required to be higher than 50% for secure standard quantum key distribution [78], which is not feasible yet with current technologies.

31 2.2. Bell s theorem 23 Figure 2.2. Schematic diagram of the ploytope in the joint probability space or alternatively in the correlation function space. The inside region of the polytope represents the accessible region of local-realistic (LR) theories and outside region contains the region of quantum mechanics (QM). Each facet of the polytope corresponds to a tight Bell inequality. A non-tight Bell inequality including bias at the boundary is represented by a line deviating from the boundary Polytope representation of Bell s theorem Bell s theorem can be represented in the space constructed by the vectors of measurement outcomes. The basis vectors correspond to either the joint probabilities or correlation functions for possible outcomes. In both spaces of joint probabilities and correlations functions the set of possible outcomes for a given measurement setting of Bell tests constitutes a convex polytope as schematically plotted in Fig. 2.2 [33, 80]. Each generator of the polytope, being an extremal point of the polytope, represents the predetermined measurement outcome called a local-realistic configuration. All interior points of the polytope are given by the convex combination of generators and they represent the accessible region of local-realistic (LR) theories associated with the probabilistic expectations of measurement outcomes. Therefore, every facet of the polytope is a boundary of halfspace characterised by a linear inequality, which we call a tight Bell inequality. There are non-tight Bell inequalities which contain the polytope in its halfspace. As a non-tight Bell inequality has interior bias at the boundary between local realistic and quantum correlations, one might call it a worse detector of non-local properties. We note that the polytope representation is

32 24 Basic Concepts useful to study Bell s theorem in complex systems, since it provides complete geometric boundary between quantum mechanics and local realism in principle for all dimensional cases. 2.3 Quantum entanglement Entanglement Entanglement is associated with the quantum correlation between two or more subsystems of a composite body. Let us consider a state that is in a Hilbert space of two qubits H a H b, Ψ ab = 1 2 ( 0 a 1 b + 1 a 0 b ). (2.20) It is impossible to determine whether the first qubit carries the value 0 or 1, and likewise for the second qubit. Only after one qubit is measured, the other qubit can be assumed to be measured in a certain state. A pure entangled state cannot be represented by a direct products of two arbitrary states as Ψ ab = ψ a ψ b, (2.21) where ψ a and ψ b are the states defined in the Hilbert space H a and H b respectively. More generally, including mixed states, an entangled quantum state is defined as a state which cannot be represented by the probability sum of direct product of density operators ˆρ i 1 and ˆρ i 2: ˆρ ab = i p i ˆρ i a ˆρ i b, p i = 1 (2.22) i

33 2.3. Quantum entanglement 25 which is called a separable state. The separable state can always be prepared in terms of local operations and classical communications (LOCC) between two separated parties, while an entangled state cannot be prepared from a separable state by LOCC. Note that entanglement decreases under LOCC and is always invariant under the local unitary transformations. Entanglement of simple models such as bipartite or tripartite 2-dimensional systems is already well understood. The entanglement criterion in a simple model is clearly defined. A quantum state with density matrix ˆρ is entangled if and only if its partial transpose has at least one negative eigenvalue [81]. This is called the Peres-Horodecki criterion [81, 82]. It was shown that this entanglement criterion is valid for 2 2 and 3 3 cases as a necessary and sufficient condition. However, it is not true for high-dimensional systems since there exist some entangled quantum states which can have positive partial transpose [82, 83]. Many theoretical proposals of entanglement criteria for various systems have been suggested and investigated [11], but we here will not go into the detail of them. Quantifying entanglement is one of the primal tasks for applications to quantum information processing. In all cases, maximally entangled states can provide ideal resources in protocols of quantum information processing. For example, a maximally entangled state is an ideal channel allowing a transfer of arbitrary quantum states with 100% fidelity in the quantum teleportation protocol [34]. Based on such a property entanglement can also be used for quantum cryptography. For example, the Ekert protocol of quantum key distribution (EK91) uses entangled pairs as the distribution channels [25], which can detect any eavesdropping attack using loophole free Bell tests.

34 26 Basic Concepts Figure 2.3. (a) Genuine and (b) non-genuine 4-dimensional entanglement model Genuine entanglement We introduce the genuine entanglement of high-dimensional or multipartite systems. Genuine d-dimensional entanglement refers to a state which is not decomposable into any sub-dimensional states. Thus a genuine d-dimensional entangled state ˆρ d can not be represented by a direct sum of any lower dimensional states, i.e. the density matrix ˆρ d cannot be written as a block-diagonal matrix, ˆρ d i C i ˆρ i where i C i = 1. We note that even a non-genuine entangled state in a d d- dimensional Hilbert space can be projected onto a maximally entangled state in a lower-dimensional Hilbert space [84]. For example, a maximally entangled state ψ 4 = and a mixed state ρ = ψ 2 ψ 2 ψ 2 ψ 2 where ψ 2 = and ψ 2 = (without normalisation factor), are different entangled states in 4 4-dimensional Hilbert space (see Fig. 2.3), but both can be mapped onto a maximally entangled state in 2 2-dimensional Hilbert space. This shows that a projection of a system to a lower-dimensional model may not preserve the whole quantum nature of the system. In other words, one can not obtain unambiguous results in the case of investigating entanglement in lowerdimensional measurements (lower than the target degrees of freedom). Similarly, genuine N-partite entanglement refers to the state in which none of the parties can be separated from any other party. Thus the genuine N-partite entangled state ˆρ N can not be represented by a product state of any lower-partite