Quantum Correlations from Black Boxes

Transcription

1 Quantum Correlations from Black Boxes GOH KOON TONG (B.Sc. (Hons.), NUS) A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Centre for Quantum Technologies National University of Singapore Supervisor: Professor Valerio Scarani Examiners: Professor Artur Ekert Professor Richard Cleve, University of Waterloo Associate Professor Jonathan Barrett, University of Oxford 2018

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3 Declaration I hereby declare that this thesis is my original work and has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Goh Koon Tong 30 April 2018 i

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5 Acknowledgements I am grateful to my supervisor, Prof. Valerio Scarani, for the opportunity to work and play in his wonderful group for half a decade since my undergraduate studies. Without his guidance, encouragement and tolerance for my quirks, I would not be able to complete this journey. During my candidature, I have made many friends who breaks the monotony of life and shared my joy and burden. I would like to take the opportunity to show my appreciation to them: in no particular order, Alexandre Roulet, Wu Xingyao, Le Phuc Thinh, Cai Yu, Stella Seah, Jean-Daniel Bancal, Jamie Sikora, Juan Miguel Arrazola, Ignatius William Primaatmaja Hadiprodjo, Le Huy Nguyen, Jędrzej Kaniewski, Andrea Coladangelo, Angeline Shu, David Luong, Stefan Nimmrichter and Aitor Villar Zafra. The administrative support from the Centre for Quantum Technologies certainly make my life during my candidature much easier and allows me to focus on research. I would like to thank Evon Tan and Siew Hoon for all the help rendered. I would also like to extend my gratitude to the Institute for Quantum Computing, University of Waterloo for hosting me as an exchange student at the start of my PhD candidature. I am forever indebted to my maternal grandmother and my parents for the unconditional support and love throughout my life. Last but not least, I would like to thank my partner, Yee Ling, for always being so understanding and supporting me in all my endeavours. iii

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7 Abstract Quantum theory predicts the existence of correlations that violate local realism. In this thesis, we look at this fact from the reverse angle ("device-independent approach"): given that such non-local correlations are observed, what can we know about the quantum resources being used? It had been known for several years that some correlations are fingerprints of specific quantum states: this kind of blind tomography has been called "self-testing". There are many existing results on self-testing but they strike one as being a special feature of certain quantum states. Here we prove that every pure bipartite entangled state possesses such fingerprints, highlighting that self-testing is a generic phenomenon. Self-testing is also used in the demonstration that a proposed connection between non-locality and uncertainty relations is an artefact of cherry-picking the expression of Bell inequality equivalent under no-signalling. This is demonstrated by synthesising equivalent Bell expression that exhibits such relations from counterexamples and vice versa. Finally, we report the first comprehensive geometric study of the set of quantum correlations, bringing to the fore an unexpected complexity of features. Equipped with this new-found knowledge, we are able to resolve the enigma behind the Bell inequality for Hardy s paradox. v

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9 List of publications This thesis is based on three publications. Chapter 2 is based on: All pure bipartite entangled states can be self-tested A. Coladangelo, K. T. Goh and V. Scarani Nature Communications 8, (2017) Chapter 3 is based on: Nonlocal games and optimal steering at the boundary of the quantum set Y.-Z. Zhen, K. T. Goh, Y.-L. Zheng, W.-F. Cao, X. Wu, K. Chen and V. Scarani Physical Review A 94 (2), (2016) Chapter 4 is based on: Geometry of the set of quantum correlations K. T. Goh, J. Kaniewski, E. Wolfe, T. Vértesi, X. Wu, Y. Cai, Y.-C. Liang and V. Scarani Physical Review A 97, (2018) During his graduate studies, the author has also contributed to the following publication. Measurement-device-independent quantification of entangled for given Hilbert space dimension K. T. Goh, J.-D. Bancal and V. Scarani New Journal of Physics 18 (4), (2016) vii

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11 Contents 1 Introduction Bell nonlocality Local realism Bell experiment Bell s inequalities Bell violation and Tsirelson s bound Realisation of Bell experiments The device-independent approach Outline of the thesis Device-independent Self-testing Background Brief history of self-testing Quantum states known to self-test All pure bipartite entangled state can be self-tested Preliminaries Outline of the proof for Theorem Conjectured self-testing correlations The ideal measurements Proof of self-testing Constructing the projections and the flip operators Constructing the unitaries ix

12 CONTENTS 2.3 Further discussion Nonlocality and Uncertainty Relation Background The Oppenheim-Wehner relation Counterexamples to the OW criterion OW criterion is an artefact of No-signalling Limitations of non-local games Equivalent Bell expressions under no-signalling OW criterion does not respect no-signalling Synthesising OW criterion Case study: Ramanathan s counterexamples Case study: Tilted-CHSH inequality Multipartite OW criterion Further discussion Geometry of the Quantum Set Background Convex set and its geometry The set of correlations The no-signalling set The quantum set The local set Boundary points of the quantum set Exposed points The CHSH point Non-extremal boundary regions Region between CHSH and deterministic point Higher-dimensional flat boundary Non-exposed extremal points x

13 CONTENTS Non-exposed point in the correlator space The Hardy s paradox Further discussion Conclusions Device-independent self-testing Nonlocality and uncertainty relations Geometry of the quantum set xi

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15 Chapter 1 Introduction In this chapter, the notions and concepts that are required to appreciate the main contents of the thesis will be formally introduced here. Here, we assume that the reader possesses basic undergraduate quantum physics and linear algebra knowledge. Following which, we will briefly introduce the phenomenon of Bell nonlocality by going through the notion of local realism and the way to falsify it in experiments. The discussion on Bell nonlocality will naturally lead us to the introduction of the device-independent approach to quantum information processing. The idea of treating quantum devices as black-boxes is the basis of all the work that will be presented in this thesis. Lastly, we will go through the outline of the thesis. 1.1 Bell nonlocality Local realism At the infancy of quantum theory, many physicists were baffled by the consequences of the new physical theory, with some even challenged its validity. In particular, Einstein, Podolsky and Rosen [40] rejected quantum theory as a complete physical theory in favour of the notion of local realism. Local realism comprises of two distinct physical principles, namely locality and realism. Their definitions are as follows: 1

16 CHAPTER 1. INTRODUCTION Locality: The belief that any instantaneous influence between objects that are spatially separated is forbidden. Realism: The belief of the pre-existence of an element of physical reality that corresponds to any physical quantities whose measured values can be predicted with certainty. Einstein, Podolsky and Rosen proposed a thought experiment where different conjugate observables (i.e. position and momentum) are measured on each of an entangled pair of particles. If one measured the position of a particle, one can infer the position of the other particle exactly with certainty. By virtue of realism, there exist an element of physical reality that corresponds to the position of the other particle. Moreover, such measurement on one particle does not affect the other, by virtue of locality. At the same time, the momentum of the other particle is being measured. Hence, we can conclude that both the position and momentum of a particle can be determine precisely. However, this is not allowed by quantum theory as it violates the uncertainty relation. Hence, in order to reconcile with the notion of local realism, quantum theory was said to be incomplete and there must exist some hidden variables that determine the outcome of the measurements, which quantum theory fails to take into account for. These hidden variables are known as local hidden variables in the literature. It is termed local as the variables are supposed to adhere to the locality principle Bell experiment In the seminal paper of Bell [15], he provided an inequality which bounds the expectation value of an observable in a certain experimental setup if one assumes that local realism holds. This result enables us to falsify local realism in an experiment, known as the Bell experiment (for illustration, please refer to Fig. 1.1). The aim of a Bell experiment is to produce correlations that is incompatible with local realism 2

17 1.1. BELL NONLOCALITY i.e. to prove that the correlation cannot be explained with a local hidden variable model. We shall call this phenomenon Bell nonlocality or nonlocality 1 for short. x y Alice no signalling Bob a b Fig. 1.1: Schematic diagram of a bipartite Bell experiment: Alice and Bob are spatially separated such that the measurement events are space-like separated which will enforce the no-signalling constraint. The measurement settings chosen by Alice and Bob are denoted by x and y respectively while the measurement outcomes are denoted by a and b respectively. A generic Bell experiment involves m 2 spatially separated parties whom we will call the first two individuals as Alice and Bob by convention 2 in most literature. The i-th party will randomly pick a measurement settings out of a possible choice of n i settings to be performed on their devices. Each measurement will, in turn, be capable of outputting k i possible outcomes. We now denote such a Bell experiment setup as a [(n 1, k 1 ), (n 2, k 2 ),, (n m, k m )] Bell scenario. Definition 1.1. A [(n 1, k 1 ), (n 2, k 2 ),, (n m, k m )] Bell scenario defines a Bell experiment with m spatially separated parties. The measurement devices of the i-th party has n i measurement settings and each measurement results in k i possible outcomes. 1 In this thesis, the term nonlocality precisely means the violation of local realism rather than instantaneous causality over space as the nomenclature may suggest. 2 This naming convention stems its origin from the field of cryptography: the characters Alice and Bob was introduced by Rivest, Shamir and Adleman [80]. The character of eavesdropper Eve was later introduced by Bennett, Brassard and Robert [17]. 3

18 CHAPTER 1. INTRODUCTION Notice that our description of Bell experiment assumes that the number of possible measurement outcomes for every setting are the same for a given party. For instance, the device of the first party, Alice, outputs a result out of k 1 possible outcomes for all n 1 settings. Hence, this notation could not describe a Bell scenario where the first measurement setting of Alice outputs 2 possible outcomes while the second measurement setting outputs 3 possible outcomes. Although this is not the most general description of Bell experiment, it is more than sufficient for the purpose for this thesis. In order for a Bell experiment to be to meaningful the simplest 3 Bell scenario is given by the [(2, 2), (2, 2)] Bell scenario. That is to say that the statistics producible by a simpler Bell scenario that has lesser parties, measurement settings or outcomes will always be compatible with local realism. Hence, one cannot witness the violation of local realism in such Bell experiment. For simplicity, we shall continue our discussion considering the [(2, 2), (2, 2)] Bell scenario. We denote the measurements settings of Alice and Bob to be x, y {0, 1} respectively with measurements outcome a, b {+1, 1} respectively (the reader may refer to Fig. 1.1). In a Bell experiment, the event when Alice randomly picks her measurement setting and the event that Bob performs the measurement must be space-like separated, and vice versa (the reader may refer to Fig. 1.2 for illustration). These constraints are imposed to prevent the information of measurement settings chosen by Alice from influencing Bob s measurement outcomes and the same for Bob. Hence, this restricts the possible correlations or behaviours, described by a set of conditional probabilities {P (a, b x, y)} a,b,x,y, produced by the Bell experiment. In particular, these no-signalling constraints [12] requires that: P (a x, y) = P (a x), (1.1) P (b x, y) = P (b y). (1.2) 3 In this context, the term simplest refers to number of parties, measurement settings and outcome. A reduction in the number of parties, measurement settings and/or outcomes would constitute a simpler Bell scenario. 4

19 1.1. BELL NONLOCALITY Time a x b y Alice Bob Space Fig. 1.2: The space-time diagram of measurement events of a Bell experiment. In order to impose the no-signalling constraints, one has to choose the measurement setting such that the measurement events of other party is outside of its light-cone. The light-cone of an event e indicates the region of events that can influence e and be influenced by e. This argument is based on the theory of special relativity. Notice that given the experimental setup, the above equalities must hold. Otherwise, superluminal communication is possible. Local realism forbids instantaneous influences across space but does not limit what the speed limit of the influence is. On the contrary, the theory of special relativity has a speed limit of the influence. Since the theory of special relativity is based on the assumption that the speed of light is constant in every inertial reference frames and is the universal speed limit, it has strong predictive power about our universe and the no-signalling constraints must hold in a Bell experiment. Indeed, both the allowed correlations from a Bell experiment by local realism and quantum theory adhere to the no-signalling principle Bell s inequalities Now, we would like to formulate the set of correlations that is compatible with local realism. Recall that local realism requires the existence of a local hidden variable model to account for the observed statistics, this means that there exists a variable λ that resides in Alice s and Bob s measurement devices that instructs the device 5

20 CHAPTER 1. INTRODUCTION what output given any measurement settings at every run. Also, we allow Alice s and Bob s devices to carry infinite amount of shared randomness. Mathematically, this implies that any correlation that is compatible with local realism must take up the following form: P (a, b x, y) = dλρ(λ)p (a x, λ)p (b y, λ) (1.3) where ρ(λ) denotes the probability density function of λ. The set of correlations that can be written in the form of equation (1.3) is known as the local polytope. The local polytope can be defined as the convex hull of all local deterministic correlations, which is given by: P (a, b x, y) = δ a,f(x) δ b,g(y) (1.4) for some functions f( ) and g( ) where δ i,j is the Kronecker delta function. A polytope is a convex set with finite number of extremal points. Hence, any point or correlation that is found outside of the local polytope exhibits nonlocality. Thus, one can conclude whether or not a correlation is compatible with local realism by checking its membership in the local polytope and such problem can be phrased as a linear program. Traditionally, showing that a correlation exhibits nonlocality involves showing that it violates a Bell s inequality. Definition 1.2. A Bell s inequality is defined as a bound on the average value of an operator ˆB that is compatible with local realism in the following form: ˆB β L (1.5) where β L denotes the maximum value of the local polytope. ˆB obtained by correlations within For instance, in the [(2, 2), (2, 2)] Bell scenario, the facets of the local polytope correspond to Bell inequalities that are known as the Clauser-Horne-Shimony-Holt 6

21 1.1. BELL NONLOCALITY (CHSH) inequalities [29], given by: A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B 1 2, (1.6) together with its permutations over settings and outcomes, with the correlators A x B y := P (a = b x, y) P (a b x, y). Formally, A x B y should be written as A x B y. Since only local measurements are allowed in Bell experiments, one can define A x := Ãx 1 and B y := 1 B y, where the tilde operators Ãx and B y act on the local Hilbert spaces that correspond to the individual systems of Alice and Bob while operators A x and B y act on the Hilbert space of the joint system between Alice and Bob. Since there is a one-to-one correspondence between A x and Ãx (similarly for other parties), we can drop the tilde sign in our notation and in the presence of the tensor product sign, A x refers to Ãx (similarly for other parties). Now, we will introduce an equivalent but different way of writing down Bell inequalities. Notice that one can write all conditional probabilities P (a, b x, y) of any set of correlations {P (a, b x, y)} a,b,x,y as a vector P. Then, any linear functional that acts on P will return a real number and we define such linear functional as a Bell expression. Definition 1.3. A Bell expression, represented by co-vector B, is a linear functional acts on a set of correlations P and returns a real number i.e. B P = k where k R. As such, we can write down the CHSH inequality in this notation as: B CHSH P 2 (1.7) where the elements B CHSH (a, b, x, y) of co-vector B CHSH is defined as: B CHSH (a, b, x, y) = ab( 1) xy. (1.8) This notation of Bell inequalities will be adopted in chapter 4 but instead of 7

22 CHAPTER 1. INTRODUCTION dealing with correlations represented with 16-dimensional vectors, we will represent the correlations with 8-dimensional vectors by taking into account of the four normalisation (of probabilities) constraints and four no-signalling constraints in the [(2, 2), (2, 2)] Bell scenario Bell violation and Tsirelson s bound The aim of the Einstein, Podolsky and Rosen paper [40] is to show that quantum mechanics cannot be complete because it is not compatible with local realism. One might expect that quantum theory will, thus, predict the existence of correlations that will violate Bell s inequality. Here, we will show that this is indeed the case, by studying the CHSH inequality. Notice that for any quantum state ρ and positive operator valued measure (POVM) 4 measurements, the state can be purified and the measurements made projective by expanding to an appropriate Hilbert space dimension. This means that any correlations produced by mixed state with POVM can be reproduced by pure state and projective measurements of a higher Hilbert space dimension. Hence, for our following proof, it suffice to consider pure quantum state ψ H A H B and projective measurements Π Ax a H A and Π By b H B corresponding to Alice making measurement x and obtaining outcome a and Bob making measurement y obtaining outcome b. Next, we have to define the operators that correspond with the correlators in the Bell expression of the CHSH inequality given in inequality (1.6) which are given by: A x = Π Ax +1 Π Ax 1, (1.9) B y = Π By +1 Π By 1. (1.10) 4 In some literature, POVM measurements are also known as generalised measurements (see reference [70] subsection 2.2.6). 8

23 1.1. BELL NONLOCALITY Notice that when we square the CHSH Bell expression, we get: (A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B 1 ) 2 = 41 1 [A 0, A 1 ] [B 0, B 1 ] (1.11) Now, we have to upper bound the largest eigenvalue of the expression [A 0, A 1 ] [B 0, B 1 ]. Since the largest eigenvalue is given by the Schatten p-norm for p =, we can exploit the properties of a Schatten norm. The Schatten p-norm, denoted by p, is given by: M p := ( i s p i (M) ) 1 p (1.12) where s i (M) is the i-th largest singular value of M. In this thesis, if the number p is omitted then it is assumed that p = 2. Now, we will show the upper bound of the -norm of [A 0, A 1 ]: [A 0, A 1 ] = A 0 A 1 A 1 A 0 (1.13) A 0 A 1 + A 1 A 0 (1.14) 2 A 0 A 1 = 2 (1.15) where we invoked the triangle inequality in the first inequality and the property of sub-multiplicativity of the Schatten p-norm in the second inequality. Also, we know that the largest eigenvalue of these operators are +1 by definition. The same argument can be made to conclude that [B 0, B 1 ] = 2. Hence, we can conclude that 41 1 [A 0, A 1 ] [B 0, B 1 ] = 8. (1.16) Therefore, by virtue of the construction of Bell expression, we recovered the Tsirelson bound [27]: A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B (1.17) 9

24 CHAPTER 1. INTRODUCTION This theorem is initially proven by Tsirelson (Cirel son) in reference [27] for any POVM measurement with a different approach that is more elegant but less pedagogical. The proof that is shown above can be found in reference [85]. Definition 1.4. A Tsirelson bound is defined as a bound on the average value of an operator ˆB that is compatible with quantum theory in the following form: ˆB β Q (1.18) where β Q denotes the maximum value of ˆB obtained by quantum correlations. One can check that the Tsirelson bound given in inequality (1.17) is tight (inequality (1.17) is saturated) by choosing the following quantum state and measurements: ψ = , (1.19) A 0 = σ z, (1.20) A 1 = σ x, (1.21) B 0 = σ z + σ x 2, (1.22) B 1 = σ z σ x 2, (1.23) where σ z and σ x are the Pauli matrices. Thus, we have shown that quantum theory predicts correlations that can violate a Bell inequality. Next, we would like to comment that although the above exercise makes obtaining Tsirelson s bound for the CHSH inequality look relatively easy, obtaining one for any generic Bell inequality remains an open problem to date. It turns out that there is no simple and efficient way to check if a given correlation can be produced by quantum systems or not. According to [5], such problem could at best be computationally hard [50] and at worst, computationally undecidable [44]. However, there exists a widely used technique, developed by Navascués, Pironio 10

25 1.1. BELL NONLOCALITY and Acín (NPA), that generates a hierarchy of relaxations to the set of quantum correlations [69], with each higher hierarchy giving a closer approximation to the quantum set than the last. Despite adopting a different framework where the NPA method impose only the commutativity relations on measurements operators for local measurements performed by different parties rather than imposing that the measurements operators are given by tensor products of local measurement operators, the NPA method provides a valid relaxation to the quantum set at every hierarchy. As a relaxation to the quantum set, it provides certification for correlations that cannot be produced by quantum systems. Hence, one can always obtain an upper bound that is not guaranteed to be tight Realisation of Bell experiments It is pointless to argue if a model or theory is complete based on physical principle that seems reasonable. In science, a physical theory must be backed by experimentation. In this aspect, local realism has been falsified experimentally on many occasions. The first Bell experiment ever performed is done by Freedman and Clauser [43], which violates the CH inequality [28]. The most famous Bell experiments were done by Aspect and his colleagues as he is the first to demonstrate the violation of the CHSH inequality [7] and also the first to perform a Bell experiment such that the measurements setting were chosen after the photons left their source and before it arrived in the detector [6]. However, the experiments performed do not account for certain loopholes. The loopholes are primarily the locality loophole and the detection loophole. The locality loophole entails that the measurements made by Alice and Bob are not space-like separated (see Fig. 1.2). The detection loophole entails that, due to the imperfection of the detectors, the violation of Bell inequality has a local hidden variable model if one discard the data when the detector does not fire [73]. Many Bell experiments 11

26 CHAPTER 1. INTRODUCTION ensued, each getting closer to close the loopholes. Finally, there were three pioneering loophole-free Bell experiments that were performed by Hensen et al. [49], Giustina et al. [46]. and Shalm et al. [86], which violates the Bell inequality while closing the loopholes. Subsequently, Rosenfeld et al. [81] also perform a loophole-free Bell experiment. Hence, there are strong evidence to believe that our universe is not compatible with the physical principle of local realism. 1.2 The device-independent approach In the previous section, the reader might already notice that one is able to falsify the notion of local realism based solely on the observed correlations in a Bell experiment without specifying the underlying mechanism of the source and measurement devices. The conclusion drawn from the Bell experiment, which was designed to be theory-independent, is also device-independent. The term device-independent is first coined by Acín and his colleagues [1] when they noticed that one is able to certify security for quantum key distribution protocol without any knowledge about the inner-workings of the device. This is inspired by the E91 protocol [41] and the Barrett-Hardy-Kent protocol [11], which show that one can distribute secure keys by violating Bell s inequality. The device-independent approach is remarkable because while the security of the BB84 protocol [16] depends on the assumption that the measurements and dimension of the Hilbert space of the signal are known 5 while the device-independent approach does not require any assumption on the underlying mechanism of the quantum devices. In this approach, by imposing the no-signalling constraints, the observed correlations could provide a certificate of security. This allows non-technician of the device hardware to verify the serviceability of the quantum device. 5 In particular, the upper bounding on the Hilbert space dimension of the signal is the crucial assumption for the security of BB84 protocol [97, 47]. 12

27 1.2. THE DEVICE-INDEPENDENT APPROACH Today, the work on device-independent quantum information processing goes beyond quantum key distribution. Researchers have taken the device-independent approach to the areas of randomness expansion (from finitely many perfect random seeds) [74, 33] and amplification (from large amount of imperfect randomness) [34], entanglement certification [66, 9], Hilbert space dimension witness for different scenarios [45, 88, 87, 23], witness for irreducible Hilbert space dimension [36] and even certification of quantum states [75, 61], which can be applied to other quantum information processing applications [21, 31]. However, all device-independent certifications require the execution of a loopholefree Bell experiment, which is technically demanding, and give pessimistic bounds on the devices performance. Hence, many are turning to relaxed version of deviceindependent schemes by adding some assumptions into the analysis. For instance, many have considered assuming the characterization of the device for one party but not the other in what is known as the steering scenario [95]. Others have considered imposing the assumption on the Hilbert space dimension of the quantum state in the device [67], which is hard to enforce in practice. A more natural constraint in this flavour is to upper bound the average photon number of the system [93]. There are also schemes, inspired by the device-independent approach, that do not follow the conventional Bell scenario set up. For instance, the measurementdevice-independent quantum key distribution [56, 20] involves two parties sending quantum state to a completely untrusted measurement device in between them. The main idea is to post-select entanglement between the two parties that enables key distribution. Nonetheless, work on device-independent quantum information processing has not been abandoned as it continues to appeal to those who seek for device certification that requires minimal assumptions on its inner-workings. 13

28 CHAPTER 1. INTRODUCTION 1.3 Outline of the thesis In this thesis, we will first explore the topic of device-independent self-testing. Selftesting is essentially the certification of quantum state in a device based only on the observed correlations from a Bell experiment. We will give a basic introduction of self-testing before concentrating on the self-testing of bipartite quantum systems. Then, we will state and show the proof for our main result that all pure bipartite entangled states can be self-tested. Next, we will move to a topic of quantum foundations that involves a proposed relationship between nonlocality and uncertainty relations in quantum theory [71]. We will then present counterexamples to this relation provided by reference [78]. Subsequently, we will show, by exploiting known self-testing results, a method to recover the relation from all known counterexamples. This reveals insights about the alleged relation between nonlocality and uncertainty relations itself. Finally, we will present a detailed study on the geometry of the set of quantum correlations. This work focuses mainly on the correlations of the [(2, 2), (2, 2)] Bell scenario. This work aims to resolve enigma in the study of quantum correlations with a different perspective. Indeed, it resolves the mystery behind the absence of a Bell expression for Hardy s paradox. 14

29 Chapter 2 Device-independent Self-testing 2.1 Background Brief history of self-testing Device-independent self-testing, or just self-testing, is arguably the strictest form of device-independent certification. The term self test was coined by Mayers and Yao [61] to describe the certification of quantum states up to some local isomorphism in a black box device using only the observed correlations. This definition of self-testing that is widely used in the literature and will be the definition of self-testing that we adopt in this thesis. Definition 2.1. A correlation, or behaviour, P (a, b x, y) self-tests the quantum state ψ target if there exists a local isometry Φ( ) such that Φ( ψ ) = junk ψ target where ψ Π Ax a Π By b ψ = P (a, b x, y) for all a, b, x and y, where ψ is the measured quantum state and the projectors Π Ax a measurements made by Alice and Bob respectively. and Π By b describe the Furthermore, we define such correlation as self-testing correlation. Definition 2.2. A self-testing correlation is a quantum correlation, or behaviour, that can only be achieved by a specific quantum state up to local 15

30 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING isometry. Additionally, in many cases, we have the full rigidity statement from a quantum correlation i.e. the observed correlation self-tests both the quantum state and the measurements. Definition 2.3. A correlation, or behaviour, P (a, b x, y) self-tests the quantum state ψ target and measurements Φ( ) such that Φ(Π Ax a Π Ax a and Π By b Π By Ax b ψ ) = junk Π a Π By b ψ = P (a, b x, y) for all a, b, x and y. if there exists a local isometry By Π b ψ target where ψ Π Ax a Notice that although definition 2.3 assumes that the quantum state is bipartite, this definition of self-testing can be extended to the multi-partite cases in a straight forward manner. That being said, we will mainly focus on the self-testing of bipartite quantum states in this thesis. Even though Mayers and Yao are often credited as the pioneers of self-testing, there are several works, which precede the one of Mayers and Yao, that also played an important role in the development of this topic. Obscured in the work of Summers and Werner [91], he made a passing remark that the only quantum state that can violate the CHSH inequality is the singlet state. Independently, Popescu and Rohrlich [76] addressed the question of what quantum state can maximally violate the CHSH inequality in their joint work and arrived at the same conclusion. Nonetheless, Mayers and Yao are the first to show that one can capitalise on the intrinsic property of quantum correlations to test quantum devices in the deviceindependent framework. Interestingly, the self-testing criterion proposed by Mayers and Yao involves checking certain correlators or correlation instead of maximal Bell violation as studied by Summers, Werner, Popescu and Rohrlich. Evidently, there exists another definition of self-testing even before the term self-testing was introduced in this context: a certification of quantum state based on the observation of a maximal Bell violation. 16

31 2.1. BACKGROUND Definition 2.4. A maximal Bell violation ˆB p = b q, where b q = max p Q ˆB p, self-tests the quantum state ψ target and measurements if there exists a local isometry Φ( ) such that Φ(Π Ax a where ψ Π Ax a Π Ax a and Π By b ψ ) = junk Π By b Ax By Π a Π b ψ target Π By b ψ = P (a, b x, y) for all a, b, x and y such that ˆB p = b q. Although definitions 2.3 and 2.4 are two distinct notions of self-testing, they are closely related to each other. On the one hand, the self-testing of definition 2.4 implies that of definition 2.3. This means that if one is able to prove self-testing via an observed maximal Bell violation, the correlations that achieved the Bell violation is also a self-testing correlation. On the other hand, the converse statement is only true if the self-testing correlation is an exposed point on the set of quantum correlations. We will elaborate on the convex geometric features of the set of quantum correlations and their implications later in chapter Quantum states known to self-test At this juncture, all the self-testing related results mentioned in the previous section only applies to the singlet state. A researcher who read the Mayers-Yao result for the first time might pose the following question: is self-testing a generic feature of quantum theory or a special property of the singlet state? At that time, a reasonable approach to this question is to find other quantum states that possess correlation that self-tests the said quantum state. Indeed, many had pursued this approach and the list of known self-testable quantum states had expanded since the publication of the work by Mayers and Yao. Since then, we know that all pure entangled two-qubits states[100, 8], the three-qubits W state [99, 72] and all graph states [62, 72] are self-testable. For higher-dimensional 1 quantum states, we know that any bipartite entangled states that can be written as a tensor product of entangled two-qubits states [98, 30, 37, 63, 68, 25] can be self-tested. Also, there are strong numerical evidence to demonstrate the self-testing of the maximally 1 We refer any quantum state with local dimension larger than 2 as higher-dimensional. 17

32 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING entangled two-qutrits state [83] and the pure partially entangled two-qutrits state that maximally violates the CGLMP 3 inequality [2, 101]. Going back to the question, self-testing is definitely not an exclusive property of the singlet state. As shown above, quantum states that are non-maximally entangled or higher-dimensional can also be self-tested. This might pique the reader s curiosity to find out which quantum states can be self-tested and which ones cannot. In the next section, we will focus on the bipartite scenario and present the complete answer for that scenario. 2.2 All pure bipartite entangled state can be selftested This section is based on the publication titled All pure bipartite entangled state can be self-tested co-authored with Andrea Coladangelo and Valerio Scarani. In this section, we will prove that all pure bipartite entangled state can be selftested. Since any correlations produced by any bipartite mixed state can be reproduced by a pure state of the same dimension[88], any bipartite mixed state cannot be self-tested. Additionally, the problem of self-testing of product states is not interesting since one can always construct a local isometry on any bipartite state that result in junk 00 by appending ancilla locally and performing trivial operation on both systems. Hence, our result gives the complete solution to the problem of which bipartite quantum states are self-testable. We will now consider a [(3, d), (4, d)] Bell scenario where the measurements settings of Alice and Bob are x {0, 1, 2} and y {0, 1, 2, 3} respectively and the outputs of Alice are given by a, b {0, 1,, d 1} respectively. A convenient way to represent correlations is through correlation tables: for the [(3, d), (4, d)], we can arrange the P (a, b x, y) in twelve d d correlation tables, one for each pair of 18

33 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED measurement settings, denoted by T x,y. a\b 0 1 d 1 0 p(0, 0 x, y) p(0, 1 x, y) p(0, d 1 x, y) T x,y := 1 p(1, 0 x, y) p(1, 1 x, y) p(1, d 1 x, y) (2.1) d 1 p(d 1, 0 x, y) p(d 1, 1 x, y) p(d 1, d 1 x, y) In the device-independent approach, the dimensionality of the measured system is not bounded a priori. Hence, the measurements made on the system can be assumed to be projective, by invoking Naimark s theorem. Hence, we are able to present any quantum strategies of Alice and Bob as a triple {Π Ax a projection Π Ax a x, and likewise for Π By b } a, {Π By b } b, ψ, where the corresponds to Alice obtaining outcome a on measurement setting on Bob s side. We do not have to assume that the unknown joint quantum state shared by Alice and Bob is pure, but rather we take it as such for ease of exposition, and we denote it by ψ. It is then easy to see that our proof holds through in the same way for a general joint quantum state ρ. Here, we will reiterate that our objective is to self-test all bipartite quantum states, and using the Schmidt decomposition, this reduces to self-testing an arbitrary bipartite quantum state of the form d 1 ψ target := c i ii (2.2) i=0 where 0 < c i < 1 for all i and d 1 i=0 c2 i = 1 for any dimension d 2. Note that since every pure bipartite quantum states with the same Schmidt coefficients c i are equivalent up to a local unitary (a change in local basis), they are, as well, equivalent up to local isometry. Hence, the above expression encompasses all pure bipartite quantum states. 19

34 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING Preliminaries Our work is heavily inspired by the work done by Yang and Navascués [100] that claimed to have proven the self-testing of all pure entangled two-qubits states and all bipartite maximally entangled states via the maximal violation of the tilted CHSH inequality[4]. However, a subsequent work by Bamps and Pironio [8] pointed out that there are non-trivial errors in the proof in the self-testing of all pure entangled two-qubits states by Yang and Navascués. Fortunately, the result still stands as Bamps and Pironio furnished the corrected proof in their work. Furthermore, modulo the typographical errors, the local isometry constructed by Yang and Navascués in their proof for self-testing of all bipartite maximally entangled states is invalid. This is due to the fact that the components that made up the supposed isometry cannot be proven to be unitary. Nonetheless, the scientific value of their work cannot be understated as it provides the necessary foundation for future works. Firstly, we will briefly introduce the tilted CHSH inequality, which will be a building block for our self-testing correlations. Let A 0, A 1, B 0, B 1 be ±1 observable. For any observable X, let X denotes its expectation. The term tilted CHSH inequality refers to a one-parameter family of Bell inequalities, which extends from the CHSH inequality with the following form [4]: α A 0 + A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B α (2.3) where α [0, 2), which holds when the correlation is compatible with local realism. The maximal quantum violation is given by 8 + 2α 2, and is attained by the quantum strategy A 0, A 1, B 0, B 1 are the dichotomic observables A 0 = σ Z, A 1 = σ X, B 0 = cos µσ Z + sin µσ X and B 1 = cos µσ Z sin µσ X, and the underlying joint quantum state is ψ = cos θ 00 + sin θ 11, where sin 2θ = 2, µ = arctan(sin 2θ), 4 α 4+α 2 and σ Z, σ X are usual Pauli matrices. The converse also holds, in the sense that maximal violation self-tests this state and these measurements [8]. The following Lemma from [8] will be useful later on. In what follows, a subscript 20

35 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED indicates the subsystem that an operator acts on (A for Alice and B for Bob). When it is clear from the context, we omit writing trivial identities on other subsystems: for example, we may write A 0 in place of A 0 1. Lemma 2.1. Let ψ H A H B. Let A 0, A 1 and B 0, B 1 be hermitian and unitary observables, respectively on H A and H B. Suppose that ψ αa 0 + A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B 1 ψ = 8 + 2α 2 (2.4) and let θ, µ (0, π 2 ) be such that sin 2θ = 4 α 2 4+α 2 and µ = arctan(sin(2θ)). Then, we set Z A = A 0, X A = A 1 and set Z B and X B be B 0+B 1 2 cos µ and B 0 B 1 2 sin µ respectively, but replacing all zero eigenvalues with one. Then, we define Z B = Z B Z B 1 and X B = X B X B 1. Finally, we have Z A ψ = Z B ψ (2.5) cos θx A (1 Z A ) ψ = sin θx B (1 + Z A ) ψ. (2.6) Now, we will move on to the final ingredient required by our work. In the work of Yang and Navascués, the authors provide a set of sufficient but not necessary conditions to prove that a given correlation self-tests a particular pure bipartite entangled state. Here, we state a slightly relaxed version of that lemma from Yang and Navascués, which gives a sufficient criterion for self-testing any pure bipartite entangled state. Lemma 2.2. Let ψ target = d 1 i=0 c i ii, where 0 < c i < 1 for all i and d 1 i=0 c2 i = 1. Suppose there exist unitary operators X (k) A, X(k) B and projections {P (k) A } k=0,..,d 1 and {P (k) B } k=0,..,d 1 of which {P (k) A } k=0,..,d 1 is a complete orthogonal set, while {P (k) B } k=0,..,d 1 need not be, and they satisfy the following conditions: X (k) A X(k) B P (k) A P (k) B (k) ψ = P ψ k, (2.7) B ψ = c k P (0) A ψ k (2.8) c 0 21

36 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING Then there exists a local isometry Φ such that Φ( ψ ) = junk ψ target, for some auxiliary state junk. 0i A 0 F S F R i AB 0i B 0 F S F R Fig. 2.1: The circuit diagram of the local isometry Φ( ψ ). The operation F and F are the quantum Fourier transform and inverse quantum Fourier transform respectively. Additionally, the operation S and R are unitary operations constructed from the measurement operators that will be defined in the main text. Proof. Recall that {P (k) A } is a complete orthogonal set of orthogonal projections by definition. Then, notice that for i j we have, using condition (2.7), P (i) B P (j) B ψ = P (i) B P (j) A ψ = P (j) A P (i) (k) A ψ = 0, i.e the P B are orthogonal when acting on ψ. Then, we can invoke a slight variation of the orthogonalization lemma (Lemma 21 from Kempe and Vidick [52]) to obtain projections on Bob s side that are exactly orthogonal, and have the same action on ψ. Lemma 2.3. Let ρ be positive semi-definite, residing in a finite-dimensional Hilbert space. Let P 1,.., P k be projections such that tr(p i P j P i ρ) ɛ (2.9) i j for some 0 < ɛ T r(ρ). Then there exist orthogonal projections Q 1,.., Q k such that k i=1 tr ( (P i Q i ) 2 ρ ) = O ( ) ɛ tr(ρ) 2 (2.10) 22

37 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED P (k) B This gives us a new set of orthogonal projections { ψ k. Now, we define the following operators: k=0 P (k) B (k) } such that P B ψ = d 1 Z A := ω k P (k) A, (2.11) d 1 Z B := ω k (k) P B k=0 d k =0 P (k ) B, (2.12) where ω denotes the root of unity. In particular, note that Z A and Z B are unitary. Moreover, notice that the expression at the end of Z B is given by ( 1 ) ψ = 0, again using condition (2.7). d 1 k=0 P (k) B Now, we define the local isometry Φ := (R AA R BB )( F A F B )(S AA S BB )(F A F B ) (2.13) where F is the quantum Fourier transform, F is the inverse quantum Fourier transform, R AA is defined so that φ A k A X (k) A φ A k A φ, and similarly for R BB, and S AA is defined so that φ A k A Z k A φ A k A φ, and similarly for S BB. We compute the action of Φ on ψ AB 0 A 0 B. For ease of notation, we drop the tildes from the (k) P B, while still referring to the new orthogonal projections. ψ AB 0 A 0 B F A F B 1 ψ AB k A k B (2.14) d k,k S AA S BB 1 d ( k,k j ω j P (j) A ) k ( ω j P (j ) B + 1 P (j ) B j j ) k ψ AB k A k B (2.15) = 1 ω jk ω j k P (j) A d ) B ψ AB k A k B k,k,j,j (2.16) = 1 ω jk ω j k P (j) A d ) A ψ AB k A k B k,k,j,j (2.17) = 1 ω j(k+k ) P (j) A d AB k A k B (2.18) k,k,j F A F B 1 d 2 k,k,j,l,l ω j(k+k ) ω lk ω l k 23 P (j) A ψ AB l A l B (2.19)

38 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING = 1 ω k(j l) ω k (j l ) P (j) d 2 A ψ AB l A l B (2.20) k,k,j,l,l R AA R BB = j j P (j) A ψ AB j A j B (2.21) X (j) A X(j) B P (j) A ψ AB j A j B (2.22) = j c j c 0 P (0) A ψ AB j A j B (2.23) = 1 P (0) A c ψ AB 0 j c j j A j B (2.24) = junk ψ target (2.25) where ω denotes the root of unity. It is easy to check that the whole proof above can be repeated by starting from a mixed joint state, yielding a corresponding version of the lemma that holds for a general mixed state. Other than providing the sufficient criterion for self-testing, Yang and Navascués [100] did not provide nor prove the existence of correlations from which one can construct operators satisfying the conditions of Lemma 2.2. Our main contribution is providing a set of self-testing correlations for each pure bipartite entangled state. Subsequently, we constructed some operators by taking products and linear combinations of the measurements operators. Finally, by using algebraic relations derived from Born s rule and the self-testing correlations provided, we proved that the constructed operators satisfies the conditions of Lemma 2.2 and hence, completing the proof of self-testing Outline of the proof for Theorem 1 Here, we will provide a sketch of the structure of our self-testing correlations that we will describe explicitly in the following subsection, and an intuition on why they should work. For the sake of clarity, we assume d to be even in this subsection. Nevertheless, the proof has a straightforward extension to odd d. 24

39 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED Recall that we aim to self-test the state ψ target = d 1 i=0 c i ii, where 0 < c i < 1 for all i and d 1 i=0 c2 i = 1 for any dimension d 2. The approach, inspired by Yang and Navascués [100], is to apply d-outcome measurements on Alice and Bob s systems such that, for some measurement settings, the correlation tables T x,y, as defined in (2.1), are block-diagonal with 2 2 blocks. More specifically, for measurement settings x, y {0, 1}, the 2 2 blocks will correspond to outcomes a, b respectively in {0, 1}, in {2, 3},, in {d 2, d 1}; and the idea is that the m-th 2 2 block certifies the portion c 2m 2m 2m + c 2m+1 2m + 1 2m + 1 of the target state (the even-odd pairs). For measurement settings x {0, 2}, y {2, 3}, the 2 2 blocks will correspond to outcomes a, b respectively in {1, 2}, in {3, 4},, in {d 1, 0} instead. Similarly, the idea being that the m-th block certifies the portion c 2m+1 2m + 1 2m c 2m+2 2m + 2 2m + 2 of the target state (the odd-even pairs). The reader may refer to Fig. 2.2 for an illustration of the concept. m = d 2 m =0 m =1 1 m = d 2 2 m = d c 0 00i c 1 11i c 2 22i c 3 33i c 4 44i c d 2 c d 1 m =0 m =1 m =2 m = d 2 1 Fig. 2.2: In blue, the block-diagonal correlations for measurement settings x, y {0, 1} certify the even-odd pairs, while, in red, the block-diagonal correlations for measurement settings x {0, 2}, y {2, 3} certify the odd-even pairs. As the reader might anticipate from earlier text, the 2 2 blocks in our blockdiagonal correlations will naturally correspond to ideal tilted CHSH correlations for appropriately chosen angles. As we will discuss later, this particular choice for the 2 2 blocks is not essential. In principle, any other self-testing correlations from which we can deduce 25

40 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING the existence of operators satisfying (2.62) and (2.63) could replace our self-testing correlations in our proof. That is to say that the proof presented in this thesis could also be extended to other block diagonal self-testing correlations. The sole reason for presenting only a single set of self-testing correlations for each pure bipartite entangled state is because there are no other known criterion for self-testing arbitrary partially entangled qubits other than the maximal violation of the tilted CHSH inequalities to date Conjectured self-testing correlations In order to self-test the target state ψ target = d 1 i=0 c i ii, where 0 < c i < 1 for any dimension d 2, listing the entire set of twelve correlation tables T x,y will not be necessary, but it will be sufficient to specify just the tables corresponding to measurement settings x, y {0, 1}, and those for settings x {0, 2}, y {2, 3}. We will show that any correlation satisfying these constraints self-tests ψ target using only the above-mentioned set of correlation tables. Additionally, the reader may find the description of the ideal measurements achieving these constraints, which can be found in subsection 2.2.4, helpful in visualising the ideal correlation. We will now state the constraints that we impose on the correlations in two parts (part (i) and (ii)), which are given by: (i) For x, y {0, 1}, the correlation tables are block diagonal with 2 2 blocks. The tables for measurement settings x, y {0, 1} are given in Tables 2.1 and 2.2 for even and odd d respectively. The 2 2 blocks C x,y,m are given by (c 2 2m + c 2 2m+1) C ideal x,y,θ m where the C ideal x,y,θ m are the 2 2 correlation tables which correspond to the maximal violation of the tilted-chsh inequality which selftests the state cos (θ m ) 00 +sin (θ m ) 11, where θ m := arctan ( c 2m+1 ) c 2m (0, π ). 2 They are given precisely in Tables , with µ m := arctan (sin (2θ m )). 26

41 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED Table 2.1: T x,y for x, y {0, 1} for even values of d 2 a\b d 2 d C x,y,m= C x,y,m= d d C x,y,m= d 2 1 Table 2.2: T x,y for x, y {0, 1} for odd values of d 3 a\b d 3 d 2 d C x,y,m= C x,y,m= d C x,y,m= d 3 2 d d c 2 d 1 27

42 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING Table 2.3: 2 2 block correlation table C x=0,y=0,m and C x=0,y=1,m a\b 2m 2m+1 2m c 2 2m cos 2 ( µm 2 ) c2 2m sin 2 ( µm 2 ) 2m+1 c 2 2m+1 sin 2 ( µm 2 ) c2 2m+1 cos 2 ( µm 2 ) Table 2.4: 2 2 block correlation table C x=1,y=0,m a\b 2m 2m+1 2m 1 2 (c 2m cos ( µm 2 ) + c 2m+1 sin ( µm 2 ))2 1 2 (c 2m+1 cos ( µm 2 ) c 2m sin ( µm 2 ))2 2m (c 2m cos ( µm 2 ) c 2m+1 sin ( µm 2 ))2 1 2 (c 2m+1 cos ( µm 2 ) + c 2m sin ( µm 2 ))2 Table 2.5: 2 2 block correlation table C x=1,y=1,m a\b 2m 2m+1 2m 1 2 (c 2m cos ( µm 2 ) c 2m+1 sin ( µm 2 ))2 1 2 (c 2m+1 cos ( µm 2 ) + c 2m sin ( µm 2 ))2 2m (c 2m cos ( µm 2 ) + c 2m+1 sin ( µm 2 ))2 1 2 (c 2m+1 cos ( µm 2 ) c 2m sin ( µm 2 ))2 (ii) Similarly, for measurement settings x {0, 2} and y {2, 3} the correlation tables T x,y are also block-diagonal, but shifted down appropriately by one measurement outcome. The 2 2 blocks are D x,y,m (corresponding to outcomes 2m + 1 and 2m + 2) for x {0, 2} and y {2, 3}, defined as D x,y,m := (c 2 2m+1 + c 2 2m+2) Cf(x),g(y);θ ideal, where m θ m := arctan ( c 2m+2 ) c 2m+1 (0, π), and f(0) = 2 0, f(2) = 1, g(2) = 0, g(3) = 1. The correlations, T x,y, for x {0, 2} and y {2, 3} are given in Tables 2.6 to 2.10 where µ m := arctan(sin(2θ m)). 28

43 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED Table 2.6: T x,y for x {0, 2} and y {2, 3}, for even values of d 2 a\b d D x,y,m= D x,y,m= d D x,y,m= d 2 1 Table 2.7: T x,y for x {0, 2} and y {2, 3}, for odd values of d 3 a\b d 2 d D x,y,m= D x,y,m= d D x,y,m= d 3 2 d c

44 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING Table 2.8: 2 2 block correlation table D x=0,y=2,m and D x=0,y=3,m a\b 2m+1 2m+2 2m+1 c 2 2m+1 cos 2 ( µ m 2 ) c 2 2m+1 sin 2 ( µ m 2 ) 2m+2 c 2 2m+2 sin 2 ( µ m 2 ) c 2 2m+2 cos 2 ( µ m 2 ) Table 2.9: 2 2 block correlation table D x=2,y=2,m a\b 2m+1 2m+2 2m (c 2m+1 cos ( µ m 2 ) + c 2m+2 sin ( µ m 2 )) (c 2m+2 cos ( µ m 2 ) c 2m+1 sin ( µ m 2 )) 2 2m (c 2m+1 cos ( µ m 2 ) c 2m+2 sin ( µ m 2 )) (c 2m+2 cos ( µ m 2 ) + c 2m+1 sin ( µ m 2 )) 2 Table 2.10: 2 2 block correlation table D x=2,y=3,m a\b 2m+1 2m+2 2m (c 2m+1 cos ( µ m 2 ) c 2m+2 sin ( µ m 2 )) (c 2m+2 cos ( µ m 2 ) + c 2m+1 sin ( µ m 2 )) 2 2m (c 2m+1 cos ( µ m 2 ) + c 2m+2 sin ( µ m 2 )) (c 2m+2 cos ( µ m 2 ) c 2m+1 sin ( µ m 2 )) The ideal measurements Having introduced the required constraints on the correlation for self-testing, we will now provide the ideal measurements on the quantum state ψ target = d 1 i=0 c i ii that will produce the ideal correlations, which satisfy these constraints. Let σ Z and σ X be the usual Pauli matrices. For a single-qubit observable A, we denote by [A] m the observable defined with respect to the basis { 2m mod d, (2m+ 1) mod d }. For instance, [σ Z ] m = 2m 2m 2m + 1 2m + 1. Similarly, 30

45 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED we denote [A] m as the observable defined with respect to the basis { (2m + 1) mod d, (2m + 2) mod d }. We use the notation A i to denote the direct sum of observables A i. For x = 0: Alice measures in the computational or the canonical basis (i.e. in the basis { 0, 1,, d 1 }). For x = 1 and x = 2: for even values of d, she measures in the eigenbases of observables d 2 1 m=0[σ X ] m and d 2 1 m=0[σ X ] m respectively, with the natural assignments of d measurement outcomes; for d odd, she measures in the eigenbases of observables d m=0 [σ X ] m d 1 d 1 and 0 0 d m=0 [σ X ] m respectively. In a similar way, for y = 0 and y = 1: for even values of d, Bob measures in the eigenbases of d 2 1 m=0[cos (µ m )σ Z +sin (µ m )σ X ] m and d 2 1 m=0[cos (µ m )σ Z sin (µ m )σ X ] m respectively, with the natural assignments of d measurement outcomes, where here µ m = arctan(sin(2θ m )) and θ m = arctan( c 2m+1 c 2m ); for odd values of d, he measures in the eigenbases of d m=0 [cos (µ m )σ Z + sin (µ m )σ X ] m d 1 d 1 and d m=0 [cos (µ m )σ Z sin (µ m )σ X ] m d 1 d 1 respectively. For y = 2 and y = 3: for even values of d, Bob measures in the eigenbases of d 2 m=0[cos 1 (µ m)σ Z +sin (µ m)σ X ] m and d 2 m=0[cos 1 (µ m)σ Z sin (µ m)σ X ] m respectively, where µ m = arctan(sin(2θ m)) and θ m = arctan( c 2m+2 c 2m+1 ); for odd values of d, he measures in the eigenbases of 0 0 d m=0 [cos (µ m)σ Z +sin (µ m)σ X ] m and 0 0 d m=0 [cos (µ m)σ Z sin (µ m)σ X ] m respectively Proof of self-testing This subsection is dedicated entirely to proving the following theorem: Theorem 1. For every bipartite entangled state of qudits ψ target, there exist correlations in the set of quantum correlations such that, when reproduced by Alice and Bob through local measurements on a joint state ρ, imply the existence of a local isometry Φ such that Φ(ρ) = ρ junk ψ target ψ target, where ρ junk is some auxiliary state. In essence, bulk of the proof involves working towards constructing operators that 31

46 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING satisfy the sufficient conditions given by Lemma 2.2. This means constructing the appropriate projections P (k) A, P (k) B and unitaries denoted by X(k) A, X(k) B. We will begin our proof by constructing the projections and also a certain type of unitary operators that will be known as the flip operators denoted by X u A,m, X u A,m. we will obtain the desired unitaries X (k) A, X(k) B After which, by taking appropriate alternating products of the flip operators. Finally, in the following subsection, we will argue that the same local isometry given by Lemma 2.2 works also to self-test the ideal measurements from subsection Constructing the projections and the flip operators The reader may recall that we denote Π Ax a as the projection corresponding to Alice obtaining outcome a on measurement setting x, and similarly for the Π By b on Bob s side. We will first derive the consequences that follow from the constraints in part (i) given in subsection 2.2.3, which we imposed on the correlations. The constraints contain in part (ii) given in subsection will bear similar implications. Now, we define the following operators: Â x,m = Π Ax 2m Π Ax 2m+1, (2.26) ˆB y,m = Π By 2m Π By 2m+1 (2.27) for x, y {0, 1}. Notice that by squaring these operators, we will obtain the following expressions: (Âx,m) 2 = Π Ax 2m + Π Ax 2m+1 := 1 Ax m, (2.28) ( ˆB y,m ) 2 = Π By 2m + Π By 2m+1 := 1 By m. (2.29) Also, one can deduce the norms 1 Ax m ψ and 1 By m ψ with the following com- 32

47 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED putation: 1 Ax m ψ = ψ (Π A 0 2m + Π Ax 2m+1) 2 ψ (2.30) = ψ Π Ax 2m + Π Ax 2m+1 ψ (2.31) = ψ (Π Ax 2m + Π Ax 2m+1) d 1 b=0 ΠBy b ψ (2.32) = c 2 2m + c 2 2m+1 (2.33) where we invoke the orthogonality of projectors to obtain the second equality, the completeness relation for the third equality and the constraints on the correlations (given in subsection 2.2.3) for the last equality. Together with the similar calculations for Bob, we can deduce that 1 Ax m ψ = 1 By m ψ = c 2 2m + c 2 2m+1 x, y {0, 1}. (2.34) Moreover, notice that ψ 1 Ax m 1 By m ψ = c 2 2m + c 2 2m+1 = 1 Ax m ψ 1 By m ψ. Hence, by using the saturation of Cauchy-Schwarz inequality, equation (2.34) and the fact that the expectation value of a projector must be non-negative, we can conclude 1 Ax m ψ = 1 By m ψ x, y {0, 1}. (2.35) By design of the structure of the ideal correlations, the correlations itself would imply that ψ α m  0,m + Â0,m ˆB 0,m + Â0,m ˆB 1,m + Â1,m ˆB 0,m Â1,m ˆB 1,m ψ = 8 + 2α 2 m (c 2 2m + c 2 2m+1) (2.36) 2 where α m =. As it stands, this does not constitute a maximal violation 1+2 tan 2 (2θ m) of the tilted CHSH inequality (since ψ has unit norm). As such, no conclusion can be drawn from equation (2.36). However, we can circumvent this problem by 33

48 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING defining the following normalised state: ψ m = 1 A 0 m ψ. (2.37) c 2 2m + c 2 2m+1 Notice that equation (2.35) implies that Âi,m ψ = and ˆB y,m ψ = equation (2.36) gives us: ˆB y,m 1 By m ψ = ˆB y,m 1 A 0 m ψ. Âx,m1 Ax m ψ = Âx,m1 A 0 m ψ, Then, inserting these relations into ψ m α m  0,m +Â0,m ˆB 0,m +Â0,m ˆB 1,m +Â1,m ˆB 0,m Â1,m ˆB 1,m ψ m = 8 + 2α 2 m (2.38) However, the operators Âx,m and ˆB y,m are not unitary. Therefore, we define the unitary operators that will replace those in equation (2.38) to be:  u x,m := 1 1 Ax m + Âx,m, (2.39) ˆB u y,m := 1 1 By m + ˆB y,m. (2.40) For the time being, we will denote unitary operators with the superscript u to differentiate them from other operators to avoid confusion. Then, by definition of ψ m and equation (2.38), we will arrive at the following expression: ψ m α m  u 0,m+Âu 0,m ˆB u 0,m+Âu 0,m ˆB u 1,m+Âu 1,m ˆB u 0,m Âu 1,m ˆB u 1,m ψ m = 8 + 2α 2 m (2.41) Now, we define the following operators Z u A,m := Âu 0,m, X u A,m := Âu 1,m. Additionally, we also define Z B,m and X B,m to be ˆB u 0,m + ˆB u 1,m 2 cos(µ m) and ˆB u 0,m ˆB u 1,m 2 sin(µ m) respectively, but with all zero eigenvalues replaced by one. Finally, we obtain the unitary operators defined by Z u B,m = Z B,m Z B,m 1 and X u B,m = X B,m X B,m 1. Then, by invoking lemma 2.1, the maximal violation of the tilted CHSH inequal- 34

49 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED ity given in equation (2.41) implies that: Z u A,m ψ m = Z u B,m ψ m (2.42) X u A,m(1 Z u A,m) ψ m = tan(θ m )X u B,m(1 + Z u A,m) ψ m (2.43) Now, we define the subspace B m = range(1 B 0 m ) + range(1 B 1 m ), and denote 1 Bm the projection onto subspace B m. Then, notice from the definition of ZB,m u, it can be written as Z u B,m = 1 1 B m + Z B,m, where Z B,m is some operator residing entirely in the subspace B m. Together with equation (2.35) and the fact that as 1 B 0 m ψ = 1 B 1 m ψ = 1 Bm ψ = 1 B i m ψ, (2.44) one can infer that Z u B,m ψ m = Z B,m ψ m = Z B,m ψ. Hence, from equation (2.42) we deduce that Â0,m ψ = Z B,m ψ. Finally, we can define the required projections P (k) A (k) and P B to be: P (2m) A := (1 A 0 m + Â0,m)/2 = Π A 0 2m, (2.45) P (2m+1) A := (1 A 0 m Â0,m)/2 = Π A 0 2m+1, (2.46) P (2m) B := (1 Bm + Z B,m )/2, (2.47) P (2m+1) B := (1 Bm Z B,m )/2. (2.48) Note that the above expressions that define P (2m) B and P (2m+1) B are indeed projections since the eigenvalues of Z B,m are ±1 in the subspace B m, and is zero outside. At this point, we would like to check that the projections defined above adhere to 35

50 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING one of the two criteria for self-testing (see equation (2.7)). P (k) A ψ = (1A 0 m + ( 1) k  0,m )/2 ψ = (1 B 0 m + ( 1) k  0,m )/2 ψ = (1 Bm + ( 1) k ZB,m )/2 ψ = P (k) B ψ (2.49) Hence, the only remaining task in this proof is to construct the unitaries X (k) A X (k) B that is compatible with the other criterion given by equation (2.8). and We will now proceed with our proof by deriving more algebraic relations between the projections and the unitary operators defined above. Notice that (1 + ( 1) k ZA,m u ) ψ m = (1 A 0 m + ( 1) k  0,m ) ψ m = (1 A 0 m plugging this relation into equation (2.43), we will arrive at: + ( 1) k  0,m ) ψ = P (k) ψ. By A XA,mP u (2m+1) A ψ = tan(θ m )XB,mP u (2m) A ψ = c 2m+1 c 2m X u B,mP (2m) A ψ, (2.50) which will come in handy later in the proof. Up to this point, we have only dealt with the part of correlations that serve to certify the even-odd pairs. Now, we turn to the constraints on our correlations that we imposed in item (ii) of subsection that will cover the certification of the odd-even pairs. These constraints bear similar implications to the ones we have just derived. We can similarly define the operators  0,m = Π 2m+1 A 0 Π 2m+2 A 0,  1,m = Π 2m+1 B 3 Π 2m+2 B 3 ( x,m A 2 ) 2 and 1 B y Π 2m+2 A ˆB 2 0,m = Π 2m+1 B 2 Π 2m+2 B 2, ˆB 1,m = Π 2m+1, and 1 A x m = m = ( 2. ˆB y,m) Using the argument employed earlier and following the same procedure, we can analogously construct unitary operators Z u A,m, X u A,m, Z u B,m and X u B,m from operators  x,m and ˆB y,m, which obeys the following relations: 36

51 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED Z A,m ψ m = Z B,m ψ m, (2.51) X A,m(1 A 0 m Z A,m) ψ m = tan(θ m)x B,m(1 A 0 m + Z A,m) ψ m, (2.52) where ψ m = 1 A 0 m ψ. Taking the same steps as before, we can deduce c 2 2m+1 +c 2 2m+2 that: X u A,m P (2m+2) A ψ = tan(θ m)x u B,m P (2m+1) A ψ = c 2m+2 c 2m+1 X u B,m P (2m+1) A ψ. (2.53) Constructing the unitaries From this point onwards, we will make a few notation changes for convenience and clarity. Since the operators we will be dealing with are unitary (except the projection operations P (k) A (k) and P ), we will drop the superscript u from the unitary operators B X u A/B,m,X u A/B,m defined in equations (2.43) and (2.53). We will also rename X u A/B,m as Y A/B,m to avoid confusion. Now, let us rewrite equations (2.50) and (2.53) in the new notation: X A,m P 2m+1 A ψ = c 2m+1 X B,m PA 2m ψ, (2.54) c 2m Y A,m P 2m+2 A ψ = c 2m+2 c 2m+1 Y B,m P 2m+1 A ψ. (2.55) Recall that we ultimately wish to produce unitary operators satisfying condition (2.8) from Lemma 2.2. The operators X A/B,m and Y A/B,m can be can be intuitively thought of as flip operators, in the sense that when X A,m acts on P (2m+1) A ψ (which is equal to P (2m+1) B ψ when condition (2.7) is satisfied), it turns into X B,m P (2m) A ψ, up to a constant factor. On the other hand, the flip operator Y A,m will turn P (2m) A ψ into Y B,m P (2m 1) A ψ, up to a constant factor. The idea is that to construct appropriate alternating products of these unitary flip operators, these products will turn P (i) A ψ into c i c 0 (X (i) B ) P (0) A ψ, which is precisely the behaviour required from condi- 37

52 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING tion (2.8) of Lemma 2.2. At this point, we have already shown, in (2.49), that the P (k) A/B, as defined in the previous subsection, satisfy P (k) A from Lemma 2.2, with the P (k) A projections. (k) ψ = P ψ for k = 0,.., d 1, i.e. condition (2.7) B forming, by definition, a complete set of orthogonal Finally, we are ready to define the unitary operators X (k) A/B as follows: X (k) A = 1, if k = 0 X A,0 Y A,0 X A,1 Y A,1 X A,m 1 Y A,m 1 X A,m if k = 2m + 1 (2.56) X A,0 Y A,0 X A,1 Y A,1 X A,m 1 Y A,m 1, if k = 2m and X (k) B = 1, if k = 0 X B,0 Y B,0 X B,1 Y B,1 X B,m 1 Y B,m 1 X B,m if k = 2m + 1 (2.57) X B,0 Y B,0 X B,1 Y B,1 X B,m 1 Y B,m 1, if k = 2m Note that X (k) A and X(k) B are unitary since they are product of unitaries. Finally we check that condition (2.8) holds, namely X (k) A P (k) A ψ = c k (X (k) c 0 B ) P (0) A ψ. (2.58) For the case of k = 0, we have: X (0) A P (0) A (0) ψ = 1P A ψ = c 0 c 0 X (0) B P (0) A ψ. (2.59) For the case of k = 2m + 1, we have: X (k) A P (k) A ψ =X A,0Y A,0 X A,1 Y A,1 X A,m 1 Y A,m 1 X A,m P (2m+1) ψ 38 A

53 2.2. ALL PURE BIPARTITE ENTANGLED STATE CAN BE SELF-TESTED (2.54) = c 2m+1 c 2m = c 2m+1 c 2m (2.55) = c 2m+1 c 2m = c 2m+1 c 2m = X A,0 Y A,0 X A,1 Y A,1 X A,m 1 Y A,m 1 X B,m P (2m) A ψ X B,m X A,0 Y A,0 X A,1 Y A,1 X A,m 1 Y A,m 1 P (2m) A ψ c 2m X B,m X A,0 Y A,0 X A,1 Y A,1 X A,m 1 Y B,m 1 P (2m 1) A ψ c 2m 1 c 2m X B,m Y B,m 1 X A,0 Y A,0 X A,1 Y A,1 X A,m 1 P (2m 1) A ψ c 2m 1 = c 2m+1 c 2m c2 c1 X B,m Y B,m 1 X B,m 1 c 2m c 2m 1 c 1 c 0 Y B,1 X B,1 Y B,0 X B,0 P (0) A ψ = c 2m+1 (X (k) c 0 B ) P (0) A ψ (2.60) which indeed proves that the constructed unitary operators adhere to equation (2.58) for k = 2m + 1. For the case of k = 2m, one can verify that similar calculation as the ones above bear the same results. One can make such conclusion by noticing that the steps taken in equation (2.60) after the third equality will yield the desired results. This completes the construction of the local isometry Φ, by Lemma 2.2, which conclude the proof of Theorem 1. Now, we will emphasise that the whole proof goes through in the same way if we replace ψ with a general mixed state. In particular, one could simply replaces all equalities between vectors with equalities between density matrices. Moreover, the Euclidean inner product will also be replaced by, : L(suppρ, H A H B ) L(suppρ, H A H B ) C such that A, B := T r(ab ρ), (2.61) where suppρ = { φ H : ρ φ 0}, and L(suppρ, H A H B ) is the space of linear maps from suppρ to H A H B. Notice that the product defined above does not satisfy the symmetric property of inner products in general. Nonetheless, Cauchy- Schwarz still holds on instances that satisfy the symmetry property, in particular when A and B commute. 39

54 CHAPTER 2. DEVICE-INDEPENDENT SELF-TESTING For instance, we would replace the expression ψ 1 A i m 1 B j m ψ, after equation (2.34), with 1 A i m suppρ, 1 B j m suppρ = T r(1 A i m suppρ 1 B j m suppρ ρ), and deduce, through Cauchy- Schwarz, that 1 A i m suppρ = 1 B j m suppρ. Finally, we would also like to remark that Lemma 2.1, from Bamps and Pironio [8], as well as Lemma 2.2, hold analogously in the corresponding mixed state form. 2.3 Further discussion a b d 1 Legend C x,y,m.... D x,y,m d 1 Fig. 2.3: Block-diagonal structure of the correlation tables In our proof, we described explicit self-testing correlations for the 2 2 blocks, in Tables and However, we would like to remark that this is not the only choice of correlations that can be made to self-test all bipartite entangled states. In fact, as a natural consequence of our work, it is the case that any block-diagonal correlations (as in Fig. 2.3) would suffice as long as the 2 2 un-normalised" correlations C x,y,m and D x,y,m imply the existence of reflections Z A, X A on Alice s 40

55 2.3. FURTHER DISCUSSION side and Z B, X B on Bob s side such that Z A ψ = Z B ψ (2.62) X A (1 Z A ) ψ = tan (θ)x B (1 + Z A ) ψ (2.63) for appropriate angles θ. For instance, in order to self-test bipartite maximally entangled states of any dimension d, we can invoke any correlation in the class given by Wang, Wu and Scarani [94] where A 0 ψ = B 0 ψ (in the notation of reference [94], α 00 = 0). These correlations satisfy equations (2.62) and (2.63) for tan θ = 1. Thus, they can be used to self-test the maximally entangled pair of qudits, for any dimension d, as is suggested by Yang and Navascués [100] up to a typographical error in the range of angles allowed. For these correlations, notice that for x = 0, y = 0, the correlation table is diagonal. Hence, we can drop Bob s fourth measurement setting because a diagonal correlation can fulfil its role as both C x,y,m and D x,y,m. Thus, one can self-test maximally entangled states of arbitrary dimension with a [(3, d), (3, d)] Bell scenario. 41

56

57 Chapter 3 Nonlocality and Uncertainty Relation 3.1 Background A remarkable open problem in the black-box approach to quantum theory is the absence of a characterisation of the set of quantum correlations with physical principles. On one hand, quantum theory predicts the existence of all local correlations and even correlations that is incompatible with local realism. This implies that the local polytope is a subset of the quantum set. On the other hand, the set of correlations that adhere to the physical principle of no-signalling, or the no-signalling polytope, is a superset of the quantum set. Many have wondered about why the quantum set does not extend to the entire no-signalling polytope? What are the other physical principles imposed on quantum theory that forbid the existence of correlations outside of the quantum set? Many attempts have been made in the past, but none manage to surmount this arduous challenge to date. In one such attempt, Oppenheim and Wehner [71] proposed that the strength of nonlocality, measured by the maximal Bell violation, in quantum theory is limited by the uncertainty principle. They showed that the proposed relation between 43

58 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION nonlocality and the uncertainty principle is plausible by proving that the relation holds for all XOR games 1, which includes the renowned CHSH game. If we have the maximal Bell violation for quantum theory for any Bell inequality, we can recover the set of quantum correlations. However, Oppenheim and Wehner conceded that it is not known if this relation holds for all non-local games in quantum theory. At this point, the reader may be confused with the supposed interchangeable use of Bell inequalities and non-local games. However, this is not true as it will be clear later that it is not sufficient to consider only non-local games in this topic. One has to consider Bell inequalities of different Bell expressions for reasons that will become clear later. Prior to the main result of this chapter, we will review the work of Oppenheim and Wehner as well as the alleged counterexamples to their work provided by Ramanathan, Goyeneche, Mironowicz and Horodecki [78]. In the main part of this chapter, we will demonstrate the relation between Bell inequalities and non-local game. Using explicit example, we will show the restrictive definition of non-local game could not capture certain features of non-locality, which a generic Bell expression could. Then, we will define the class of distinct Bell expressions that are equivalent under the no-signalling principle. Thereafter, we will show that the validity OW criterion varies even for equivalent Bell expressions under the no-signalling principle. In order to drive the point, we will write down an explicit counterexample to the OW criterion that is equivalent to the CHSH inequality, which was shown to obey the OW criterion by reference [71]. Furthermore, we will write down a Bell expression, which obeys the OW criterion, for each of the counterexamples provided by reference [78]. Finally, we will conclude that the criterion is merely a feature that can be derived due to the equivalence of non-local games under the no-signalling principle. 1 XOR games are non-local games which winning conditions are imposed on the output of the XOR logical operation on the measurement outcomes of Alice and Bob 44

59 3.1. BACKGROUND The Oppenheim-Wehner relation In a non-local game, we have a bipartite Bell scenario where the spatially separated Alice and Bob receive binary inputs x and y respectively from a referee. In return, they each send the referee a binary output a and b respectively. Then, the referee will feed the bits a, b, x and y into a Boolean function, G abxy {0, 1}, which outputs a bit. If the output of G abxy is the bit 1 Alice and Bob win the game. Otherwise, they lose the game. Clearly, the Boolean function G abxy defines a particular nonlocal game we denote as G. The winning probability, ω(p, G) of the non-local game G given a strategy P (a, b x, y) or its short hand P is given by: ω(p, G) := xy π(xy) ab G abxy P (a, b x, y) (3.1) where π(xy) is the joint probability distribution of the input variables x and y. We denote the maximum winning probability. Notice that one can also derive a Bell inequality from a non-local game. We now denote ω L (G) as the maximum winning probability of non-local game G over all local strategies. Thus, we arrive at a Bell inequality: π(xy)g abxy P (a, b x, y) ω L (G) (3.2) xyab where terms π(xy)g abxy play the role of the coefficients of a Bell expression. Now, we denote ω Q (G) as the maximum winning probability of non-local game G over all quantum strategies in order to construct the following Tsirelson-type bound: π(xy)g abxy P (a, b x, y) ω Q (G) (3.3) xyab and this inequality can be re-written as: π(x)p (a x) xa yb π(y x)g abxy P (b y, x, a) ω Q (G). (3.4) 45

60 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION The purpose of rewriting the expression of the winning probability in this particular form is to introduce the fine-grained uncertainty relations. For a given (x, a) pair, the fine-grain uncertainty relations for some conditional input probability distributions π(y x) and quantum measurements on Bob s system, Π By b, is defined as: π(y x)g abxy P (b y, x, a) λ x,a x, a (3.5) yb where λ x,a can be determined by the largest eigenvalue of yb π(y x)g abxyπ By b. It is named an uncertainty relation because the set of inequalities provides an upper bound to some linear combinations of conditional probabilities. To a certain extent, the saturation of these inequalities implies maximal certainty for the output b when applying the measurements Π By b on Bob s quantum state that is prepared by Alice s measurement. This prepared state or conditional state, which we will denote as ρ B x,a, is given by: ρ B x,a := Tr [ ] A (Π A x a 1) ρ AB P (a x) (3.6) where ρ AB is the shared quantum state between Alice and Bob. The prepared state that saturates the fine-grained uncertainty relation is also known as the maximal certain state as it gives rise to maximal certainty for given Bob s measurements Π By b. In the language of steering, the set of {P (a x), ρ B x,a } x,a, where ρ B x,a is the unnormalised state P (a x)ρ B x,a, is known as an assemblage. The reader may refer to Fig. 3.1 for illustration of the idea of steering. Looking back at the left hand side of inequality (3.4), if Bob performs the optimum measurements for the particular non-local game, then the winning probability will only depend on the possible assemblages. Hence, the quantum optimal winning probability ω Q (G) will be attained with some optimal assemblage. Oppenheim and Wehner claim that for all XOR games and other games where the optimal measurements are known, the optimal assemblage that gives rise to the 46

61 3.1. BACKGROUND x Steers to a { B x,a,p(a x)} x,a Fig. 3.1: The schematic diagram of a steering scenario: Steering involves the usual Bell scenario except that one of the devices is fully characterised i.e. the quantum state in the device can be fully determined. By performing measurements on her device, Alice could remotely steers the state in Bob s device. In a way, Alice prepares the state ρ B x,a on Bob s device when she makes measurements x and obtains outcome a. quantum optimal winning probability ω Q (G) corresponds to the maximal certain states. Hence, the Oppenheim-Wehner (OW) criterion states that maximal nonlocality (saturation of inequality (3.4)) is a consequence of maximal certainty (saturation of inequalities (3.5)) through steering. Oppenheim and Wehner also admitted that it is an open problem whether this relation holds for all non-local games in quantum theory. If the OW criterion holds for any non-local games in quantum theory, then necessarily, any optimal quantum strategies (quantum state and measurements) for any non-local games must saturate all the fine-grained uncertainty relations Counterexamples to the OW criterion In the previous subsection, we have established that the OW criterion requires the saturation of all the fine-grained uncertainty relations derived from a Bell inequality whenever the said Bell inequality is maximally violated. Thus, this criterion imposes a constraint on the value of the maximal violation of the said Bell inequal- 47

62 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION ity. However, it was not known if this criterion holds for Bell inequalities other than those associated to the XOR non-local games. Finding a Bell inequality or a non-local game that does not adhere to the above criterion would constitute as a counterexample to the OW criterion. In the work of Ramanathan, Goyeneche, Mironowicz and Horodecki [78], two counterexamples to the OW criterion were proposed. We will first establish the assumptions and notations that will be used in this chapter, followed by the introduction of these counterexamples. As mentioned earlier at inequality (3.2), any non-local game G abxy can be phrased as a Bell inequality where π(xy)g abxy is the coefficients of the Bell expression. For simplicity, we will assume that π(xy) = π(x)π(y) where π(x) = π(y) = 1 2 for the rest of this chapter since π(xy) that are not uniform is not required for our discussion. Under this assumption, the Bell inequality that is associated with the non-local game G abxy has coefficients of the Bell expression given by G abxy itself. The Bell inequality is given as follows: B(P ) := abxy G abxy P (ab xy) β L (3.7) where β L is the maximum value of B(P ) compatible with local realism and the value of B(P ) is four times the winning probability when adopting strategy P. form: For convenience, we will represent a Bell expression B in the following tabular G 0000 G 0100 G 0001 G 0101 B := G 1000 G 1100 G 1001 G 1101 G 0010 G 0110 G 0011 G (3.8) G 1010 G 1110 G 1011 G 1111 The Bell expressions of the two counterexamples to the OW criterion, denoted 48

63 3.1. BACKGROUND by B c1 and B c2, given by Ramanathan and his colleagues are as follow: B c1 = (3.9) and B c2 = (3.10). Ramanathan and his colleagues have also found the unique quantum states and measurements that gives rise to the optimal winning probabilities for both B c1 and B c2. Now, we can list the set of fine-grained uncertainty relations for both B c1 and B c2. The fine-grained uncertainty relations, now written as G abxy P (b y, x, a) λ (x,a) (3.11) y,b where λ x,a is the largest eigenvalue of yb G abxyπ By b, of B c1 are given by: P (0 0, 0, 0) λ c1(0,0), (3.12) P (0 1, 0, 1) λ c1(0,1), (3.13) P (1 0, 1, 0) + P (1 1, 1, 0) λ c1(1,0), (3.14) P (0 0, 1, 1) λ c1(1,1), (3.15) where λ c1(0,0) = λ c1(0,1) = λ c1(1,1) = 1 and λ c1(1,0) Also, the fine-grained 49

64 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION uncertainty relations of B c2 are given by: P (0 0, 0, 0) + P (1 1, 0, 0) λ c2(0,0), (3.16) P (1 0, 0, 1) + P (0 1, 0, 1) λ c2(0,1), (3.17) P (1 0, 1, 0) + P (1 1, 1, 0) λ c2(1,0), (3.18) P (0 0, 1, 1) λ c2(1,1), (3.19) where λ c2(0,0) = λ c2(0,1) , λ c1(1,0) and λ c1(1,1) = 1. Ramanathan and his colleagues showed that the conditional states ρ B x,a of the optimal quantum strategy for B c1 do not saturate fine-grained uncertainty relations (3.12), (3.13) and (3.15). Ramanathan and his colleagues noted that the fine-grained uncertainty relations (3.12), (3.13) and (3.15) are trivial relations, which means that they impose nothing more than the fact that probabilities must be lesser than or equal to unity. Thus, they went on to show that the OW criterion also breaks down in the case of non-trivial fine-grained uncertainty relations using the Bell expression B c1. In the case of B c1, the conditional states ρ B x,a of the optimal quantum strategy do not saturate the fine-grained uncertainty relations (3.16), (3.17) and (3.19). Hence, we can conclude that the OW criterion does not hold for all non-local games. 3.2 OW criterion is an artefact of No-signalling This section is based on the publication titled Non-local games and optimal steering at the boundary of the quantum set co-authored with Yi-Zheng Zhen, Yu-Lin Zheng, Wen-Fei Cao, Xingyao Wu, Kai Chen and Valerio Scarani. Even though the work of Ramanathan and his colleagues dealt the final blow to Oppenheim and Wehner s attempt to describe the set of quantum correlations with physical principles, they did not provide an autopsy for the OW criterion. Such information may be valuable to researchers who are working on this topic as they could apply consistency checks on their work to avoid the mistakes made by their 50

65 3.2. OW CRITERION IS AN ARTEFACT OF NO-SIGNALLING predecessors. In this section, we seek the reason why OW criterion holds for some Bell expression and fails for others. First, we will give a short remark that non-local games cannot be used to express certain Bell expressions. Hence, any property that is intrinsic to nonlocality must hold beyond non-local games. As such, our discussion must be expanded to Bell expressions. Next, we will show that the OW criterion does not respect the equivalence under no-signalling. That is to say that for a Bell expression that complies with the OW criterion, one can find an equivalent Bell expression under no-signalling principle that violates the OW criterion. Finally, we show that it is possible to synthesise the form of Bell expressions that comply with the OW criterion for those that do not. Hence, the OW criterion emerges as a coincidence when the form of the written Bell expression fulfils certain mathematical prerequisites and not a constraint on the strength of nonlocality in quantum theory imposed by uncertainty relation Limitations of non-local games In inequality (3.2), we showed that any non-local games can be written down as a Bell expression. However, in this subsection we will see that the converse is not true. We will demonstrate this relation between non-local games and Bell expression by showing an explicit example where a Bell expression cannot be written as a non-local game. Consider a physical setup of a Bell experiment of the [(2, 2), (2, 2)] Bell scenario, such that the chances of performing any measurement are unbiased i.e. π(x, y) = π(x)π(y) = 1 1. Now, recall the tilted-chsh inequality introduced in inequal- 2 2 ity (2.3) and for convenience of the reader, it will be displayed here: α A 0 + A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B α (3.20) 51

66 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION where α [0, 2). Also, notice that: A 0 = P (a = 0 x) P (a = 1 x) (3.21) = b P (0, b x, y) P (1, b x, y) y. (3.22) Since the no-signalling constraint requires equation(3.22) to hold for all y, there are two ways of writing the Bell expression in our table notation: 1 + α α B α-chsh,1 = 1 α 1 α α, (3.23) and α 1 + α B α-chsh,2 = α 1 α α. (3.24) Since for non-local games, G abxy {0, 1} are Boolean variables, notice how the values of π(x, y)g abxy are unable to represent the Bell expressions above even if we allow π(x, y) to vary. Thus, it is not sufficient to consider only non-local games if we aim unveil the intrinsic properties of nonlocality in quantum theory. On the contrary, a relation that holds for all Bell expressions would hold for all non-local game as well Equivalent Bell expressions under no-signalling Before diving into the main result, we will now introduce the transformation between Bell expressions that are equivalent under no-signalling principle. Here, we keep the 52

67 3.2. OW CRITERION IS AN ARTEFACT OF NO-SIGNALLING discussion in the [(2, 2), (2, 2)] Bell scenario, but the generalisation is obvious. In our notation, a Bell expression of a Bell inequality B is represented by the table (3.8), where G abxy R instead of {0, 1} in the case of non-local games, and any real number k can be represented by the tables k k k k k = k k 0 0 = = 0 0 k k = , (3.25) k k k k k k k k and any convex combinations of these tables due to the normalisation constraint a,b P (a, b x, y) = 1 x, y. This representation of a constant is pretty trivial and indeed the OW character 2 of a Bell expression is not changed by adding k expressed in this way. This is evident because the maximal violation of Bell inequality by quantum strategies will be shifted by a constant can be achieved with the same optimal quantum strategy. Additionally, adding k in this way will merely introduce a constant on both sides of the fine-grained uncertainty relations expressed by inequality (3.11). As such, starting from a Bell expression B, one can always construct an equivalent Bell expression B with the same OW character such that all the G abxy are non-negative by adding constants. Alternatively, if we enforce the no-signalling constraint for Bob P B (b y) := P (b x = 0, y) = P (b x = 1, y), we obtain less trivial 2 whether or not a Bell expression adhere to the OW criterion 53

68 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION representations of the same constant: k k k k k = k = 0 k 0 0 = 0 0 k 0 = k, (3.26) 0 k 0 0 k k 0 0 k 0 0 k 0 0 k k 0 0 k 0 and convex combinations thereof. Similarly, enforcing the no-signalling constraint for Alice P A (a x) := P (a x, y = 0) = P (a x, y = 1), we have the additional representations k k k k k = 0 0 k k = k k 0 0 = = , (3.27) k k k k k k k k 0 0 and convex combinations thereof. Such re-writings based on the no-signalling principle may alter the OW character of a Bell expression. This will be clearly illustrated by the CHSH game in section Having come to terms with this flexibility, it is convenient to recast the same information in difference tables denoted by D and given by D = D 00 D 01 D 10 D 11 (3.28) 54

69 3.2. OW CRITERION IS AN ARTEFACT OF NO-SIGNALLING with G 00xy G 01xy D xy = G 00xy G 10xy G 01xy G 11xy. (3.29) G 10xy G 11xy This representation is handy because the transformations allowed by no-signalling can be expressed in a simple form. Indeed, two difference tables are equivalent under no-signalling if and only if there exist some α, β, γ, δ R such that D = D + (α, β, γ, δ) where (α, β, γ, δ) is defined as: +α +β +γ +γ γ γ (α, β, γ, δ) = +α +β α β +δ +δ δ δ. (3.30) α β The above statement is true because adding of constant k = k ab P (a, b x, y) for any x, y does not change the difference between any G abxy terms of the same x, y. Additionally, performing a transformation given by k [P (0, 0 0, 0) + P (0, 1 0, 0)] = k [P (0, 0 0, 1) + P (0, 1 0, 1)] decreases the value of G 0000 G 1000 and G 0100 G 1100 by k while increases the value of G 0001 G 1001 and G 0101 G 1101 by k. In particular, if a difference table D k represents a constant k under the no- 55

70 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION signalling constraints for Alice and Bob, there must exist α, β, γ, δ such that D k = D k + (α, β, γ, δ) = (3.31) OW criterion does not respect no-signalling In the [(2, 2), (2, 2)] Bell scenario, the eminent CHSH inequality [29] is the only tight Bell inequality (facet of the local polytope). Its maximal violation self-tests the singlet state and defines a single point on the boundary of the set of quantum correlations. The basic results of the Oppenheim-Wehner paper is that the OW criterion holds for that point. However, what is not clear from the paper is that the conclusion depends on the Bell expression that is used. This observation is the basis of our work, and we will provide an explicit example now. We start with the CHSH XOR game B CHSH = 1 x,y=0 P (a x b y = xy), where P (a x b y = xy) = a P (a, b = xy a xy). It has βchsh L = 2 and βq CHSH = 2+ 2 where β L and β Q are the maximum Bell values given by local realism and quantum theory respectively. Since the optimal winning probability self-tests, the quantum state and measurements that achieve β Q can be uniquely written, up to local isometries, as Φ + = ( ) / 2, and A 0 = σ z, A 1 = σ x, B 0 = (σ z + σ x ) / 2, B 1 = (σ z σ x ) / 2 56

71 3.2. OW CRITERION IS AN ARTEFACT OF NO-SIGNALLING [75]. Now, consider the following re-writing: B CHSH = = = 2B CGLMP The Bell expression B CGLMP 2 is the CGLMP 2 in the version of Zohren and Gill [102], which was already known to be equivalent to CHSH for no-signalling correlations. Note that the CGLMP 2 Bell expression comes from the family of Bell inequalities known as the GCLMP d inequalities [51, 35], which is a set of local polytope facets of the [(2, d), (2, d)] Bell scenario [58]. Now, let us prove that the rightmost table is a complicated way of writing the constant k = 3 for no-signalling correlations. The top left block is P (0, 0 0, 0) + 2P (0, 1 0, 0) + P (1, 1 0, 0) = P A (0 0) + P B (1 0). Treating the two off-diagonal blocks similarly, we find that the table represents [P A (0 0)+P B (1 0)]+[P A (1 0)+P B (0 1)]+ [P A (1 1)+P B (0 0)]+P (0, 1 1, 1) P (1, 0 1, 1) = 2+P A (1 1)+P B (0 1)+P (0, 1 1, 1) P (1, 0 1, 1). Exploiting the same argument using no-signalling, one has P A (1 1) = P (1, 0 1, 1)+ P (1, 1 1, 1) and P B (0 1) = P (0, 0 1, 1) + P (1, 0 1, 1), which proves the claim. Alternatively, one can also prove the above-mentioned claim using the difference tables introduced earlier. Hereafter, we will leave it to the reader to check the transformations of Bell expressions under no-signalling. Following from these observations that β CGLMP 2 Q is obtained for the same point on the boundary of the set of quantum correlations that gives βq CHSH, which we said is only achievable with the quantum states and measurements written above. Then, given the operators, the bounds yb G abxyp (b y, x, a) λ CGLMP 2(x,a) of B CGLMP 2 57

72 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION are given in terms of the P (b y, x, a) by P (0 0, 0, 0) + P (1 0, 0, 0) + P (0 1, 0, 0) λ CGLMP 2(0,0), (3.32) P (1 0, 0, 1) + P (0 1, 0, 1) + P (1 1, 0, 1) λ CGLMP 2(0,1), (3.33) P (0 0, 1, 0) + P (1 1, 1, 0) λ CGLMP 2(1,0), (3.34) P (0 0, 1, 1) + P (1 0, 1, 1) = 1, (3.35) where λ CGLMP 2(0,0) = λ CGLMP 2(0,1) = 2 and λ CGLMP 2(1,0) = Notice that conditions like P (0 0, 0, 0) + P (1 0, 0, 0) = 1 hold by definition, so one could just have written the first line as P (0 1, 0, 0) λ CGLMP 2(0,0) where λ CGLMP 2(0,0) = λ CGLMP 2(0,0) 1; and similarly for the second line. However, for the conditional quantum state of the optimal quantum strategy ρ B x,a, y,b G ab0yp (b y, 0, a) = < 2 = λ GCLMP 2(0,a) for both a = 0, 1. Hence, the CGLMP 2 game B CGLMP 2 does not adhere to the OW criterion. This case study shows that different Bell expressions equivalent under no-signalling may behave differently on the OW characterisation Synthesising OW criterion In this subsection, we will discuss how to synthesise a Bell expression that satisfy the OW criterion. This will give us an insight on why the OW criterion works on certain Bell expression and not on others. First, notice that even the verification of the OW criterion for a given Bell expression is not trivial a priori. Indeed, while the property of lying on the boundary of the quantum set is determined by the Bell expression alone, checking the OW criterion involves finding the states and operators that realise the optimal quantum correlation P op. For a generic Bell expression, it is not known how to find such a quantum realisation. Even if one is found, there is no guarantee that it is the unique optimal quantum strategy. If P op could be obtained with inequivalent realisations of the state and the measurements, one would have to say whether saturation of 58

73 3.2. OW CRITERION IS AN ARTEFACT OF NO-SIGNALLING (3.11) holds for all realisations, or it is suffice that it holds for one. As it turns out, for all the cases that we explicitly studied so far, a P op on the quantum boundary can be obtained by a unique choice of the state and the measurements, up to local isometries as the maximal violation of the Bell s inequalities self-tests the state and the measurements. Having clarified how the OW criterion is going to be tested, we need to move one step back and explain how one can try and guess a candidate Bell expression that may adhere to the OW criterion, given all the freedom allowed by the equivalence under no-signalling. The heuristic method we found consists in enforcing first some necessary conditions. Indeed, it is clear that the OW criterion can only be satisfied if yb G abxyπ Bx b is diagonal in the basis which ρ B x,a is diagonal. This condition imposes several constraints on the values of G abxy, that largely restrict the candidate Bell expressions. The remaining ones can then be tested directly. Now, we show more explicitly how to implement the constraints discussed above in the case of self-testing correlations in the (2, 2, 2, 2) Bell scenario. In this scenario, we are guaranteed that the maximal Bell violation by a quantum resource can be achieved by a pure bipartite qubits state and projective measurements [59]. If the point is self-testing, then it does self-test a pure two-qubits state and those measurements. In particular, the conditional state on Bob will be a pure qubit state. Define now the unitary transformation U x,a such that U x,a ρ B x,a U x,a = 0 0. (3.36) The projectors written in the basis where the steered state is diagonal are given in 59

74 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION the following form: U x,a Π B 0 0 U x,a = U x,a Π B 0 1 U x,a = U x,a Π B 1 0 U x,a = U x,a Π B 1 1 U x,a = p 0 (x, a) q 0 (x, a) (3.37) q 0 (x, a) 1 p 0 (x, a) 1 p 0 (x, a) q 0 (x, a) (3.38) q 0 (x, a) p 0 (x, a) p 1 (x, a) q 1 (x, a) (3.39) q 1 (x, a) 1 p 1 (x, a) 1 p 1 (x, a) q 1 (x, a) (3.40) q 1 (x, a) p 1 (x, a) where p 0 (x, a), q 0 (x, a), p 1 (x, a) and q 1 (x, a) are some real numbers between 0 and 1. The necessary conditions for a Bell expression G abxy that adhere to the OW criterion are given by: q 0 (x, a)(g a0x0 G a1x0 ) + q1(x, a)(g a0x1 G a1x1 ) = 0 x, a. (3.41) In particular, for cases where the r(x, a) := q 0(x,a) q 1 (x,a) are well-defined for all x, a pairs, a Bell expression that adhere to the OW criterion must have the form: A B C r(0, 0)(A B) + C B OW = D E F r(0, 1)(D E) + F G H I r(1, 0)(G H) + I J K L r(1, 1)(J K) + L (3.42) where the capital Roman alphabet letters are free variables. In order to check whether a Bell expression B has an equivalent form under 60

75 3.2. OW CRITERION IS AN ARTEFACT OF NO-SIGNALLING no-signalling that adhere to the OW criterion, we can now take B and check if there exist an OW-game such that B B OW = k. As discussed before, this check is going to be simplest by passing it in the difference representation and using equation (3.31). Case study: Ramanathan s counterexamples Ramanathan and coworkers [78] provided two non-local games that violates the OW criterion. However, we can write down equivalent Bell expressions to these counterexamples to show that there is no fundamental differences between the optimal quantum correlations for these non-local games and those for the XOR games that can be derived from the differences in their OW characters. Rather, it is the form of the Bell expression used that determines the OW character. For the first counterexample, we write down the family of Bell expressions B c1 (Γ), given by: Γ 1 Γ 1 B c1 (Γ) = Γ 1 Γ 1 = B c1 (0) + 2Γ. (3.43) Γ 1 0 Γ Γ 1 0 Γ 1 The fact that the rightmost table is equal to k = 2Γ for no-signalling correlations can be checked using the difference representation of the tables given in equation (3.31). The bounds yb G abxyp (b y, x, a) λ (x,a) for B c1 (Γ) is given by: ΓP (0 0, 0, 0) + (1 Γ)P (1 1, 0, 0) λ c1(0,0), (1 Γ)P (1 0, 0, 1) + ΓP (0 1, 0, 1) λ c1(0,1), P (1 0, 1, 0) + P (1 1, 1, 0) λ c1(1,0), (2 Γ)P (0 0, 1, 1) + (1 Γ)P (0 1, 1, 1) λ c1(1,1), 61

76 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION In reference [78], it is proved that B c1 (0) self-tests a given two-qubits state and suitable measurements; and that the OW criterion do not hold. However, using those same state and measurements, for Γ we find numerically that the criteria hold, with λ c1(0,0) = λ c1(0,1) , λ c1(1,0) , λ c1(1,1) by: For the second counterexample, we use the family of Bell expressions B c2, given Γ Γ 0 B c2 (Γ) = Γ Γ 0 = B c2 (0) + 0 (3.44) Γ 0 0 Γ Γ 0 0 Γ because the table we added is equal to k = 0 for no-signalling correlations. The corresponding bounds (1 Γ)P (0 0, 0, 0) + (1 + Γ)P (1 1, 0, 0) λ c2(0,0), (1 + Γ)P (1 0, 0, 1) + (1 Γ)P (0 1, 0, 1) λ c2(0,1), P (1 0, 1, 0) + P (1 1, 1, 0) λ c2(1,0), (1 + Γ)P (0 0, 1, 1) + ΓP (0 1, 1, 1) λ c2(1,1), are not saturated for Γ = 0, as shown in reference [78]; but they are for Γ , in which case λ c2(0,0) = λ c2(0,1) = λ c2(1,1) , λ c2(1,0) Case study: Tilted-CHSH inequality Now, we will like to consider, again, the tilted CHSH inequalities [2] which were previously written in our table notation in inequalities (3.23) and (3.24). Notice that by adding a constant of 4 + α on both side of the inequality, one will arrive at: 62

77 3.2. OW CRITERION IS AN ARTEFACT OF NO-SIGNALLING 1 + α α 1 0 B α (Γ) = α (3.45) As mentioned in the previous chapter, the maximal quantum violation of each of these Bell inequalities self-tests a corresponding partially entangled qubits state ψ(θ) = cos θ 00 + sin θ 11 with α = 2/ tan 2 2θ, for the measurements A 0 = σ z, A 1 = σ x, B 0 = cos µσ z + sin µσ x and B 1 = cos µσ z sin µσ x where tan µ = sin 2θ [100, 8]. For this example, we work with the family of Bell expressions 1 + α α cos 2θ 0 cos 2θ B α (Γ) = Γ 0 cos 2θ 0 cos 2θ (3.46) sin 2 θ cos 2 θ sin 2 θ cos 2 θ sin 2 θ cos 2 θ sin 2 θ cos 2 θ The rightmost table is k = 2 sin 2 θ for no-signalling correlations, so the local bound is I L α (Γ) = 3 + α 2Γ sin 2 θ. The case when Γ = 0 corresponds to the table given in inequality (3.45). For the states and measurements that give rise to the maximum quantum violations, we have xa G abx1p (a x, y = 1, b) = 1 + (1 cos 4θ)/( 6 2 cos 4θ), which is strictly smaller than λ (1,b) = 1 + sin 2θ for α (0, 2] which shows that B α (0) does not adhere to the OW criterion. 2 3 cos 4θ However, Γ = 1 corresponds to a Bell expression that respects the OW criterion. 63

78 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION In this case, the bounds yb G abxyp (b y, x, a) are given by P (0 0, 0, 0) + P (0 1, 0, 0) λ α(0,0), P (1 0, 0, 1) + P (1 1, 0, 1) λ α(0,1), cos 2 θp (0 0, 1, 0) + sin 2 θp (1 1, 1, 0) λ α(1,0), sin 2 θp (1 0, 1, 1) + cos 2 θp (0 1, 1, 1) λ α(1,1), 2 with λ α(0,0) = cos 4θ 3 cos 4θ α(0,1) = 3 cos 4θ 6 2 cos 4θ and λ α(1,0) = λ α(1,1) = cos 4θ. Interestingly, even if these Bell expressions are asymmetric under the exchange between Alice and Bob, they can be used to steer in the either direction. We have just presented the steering from Alice to Bob. That from Bob to Alice, the OW criterion is given by x,a G abxyp (a x, y, b) = λ (y,b) y, b where λ (y,b) is the largest eigenvalue of yb V abxyπ x a. From the tilted CHSH inequalities, we can write down another family of Bell expressions, denoted by B α(γ), which is given by: 1 + α α 1 0 α α 0 0 B α(γ) = Γ X 1 α X 1 α X 1 X 1 (3.47) X 2 X 2 1 X 2 1 X 2 where X 1 = 2 Λ + Λ Λ + Λ, X 2 = 2Λ +Λ Λ + Λ 2 sin Λ + Λ and Λ ± = 2 2θ 1+sin. Similarly, 2 2θ 1 2 sin 2 θ± B α(γ) fulfils the OW criterion for the case Γ = 1 but not for the case of Γ = 0. In 64

79 3.2. OW CRITERION IS AN ARTEFACT OF NO-SIGNALLING the case Γ = 1, the bounds x,a G abxyp (a x, y, b) λ (y,b) are given by P (0 0, 0, 0) + (X 1 α)p (1 0, 0, 0) +P (0 1, 0, 0) + X 2 P (1 1, 0, 0) λ α(0,0), (X α)p (1 0, 0, 1) + (X 2 + 1)P (1 1, 0, 1) λ α(0,1), P (0 0, 1, 0) X 1 P (1 0, 1, 0) + P (0 1, 1, 0) +(2 X 2 )P (1 1, 1, 0) λ α(1,0), (1 X 1 )P (1 0, 1, 1) + 2P (0 1, 1, 1) +(1 X 2 )P (1 1, 1, 1) λ α(1,1). For the steering scenario of Bob to Alice, the probabilities written above are given by the notation P (a x, y, b) Multipartite OW criterion All discussions made on OW criterion in the literature focus on the bipartite Bell scenarios, which is not surprising because the idea of steering is somehow naturally bipartite. However, a multipartite generalisation of steering has been introduced. In this subsection, we will introduce a natural way to writing down a possible extension of OW criterion to the multipartite scenario. Following the work of Cavalcanti and his colleagues [24], in the tripartite Bell scenario, one can distinguish two types of steering: (i) one black box that steers to two characterised devices and (ii) two black boxes that steer to one characterised device. The reader may refer to Fig 3.2 for illustration of the idea. Each type of steering would give rise to different sets of OW criterion for a Bell expression defined by G abcxyz, namely if the observed correlations violates Mermin 65

80 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION (i) x Steers to x a y { BC x,a,p(a x)} x,a (ii) Steers to a b { C x,y,a,b,p(a, b x, y)} x,y,a,b Fig. 3.2: Schematic diagrams of steering for multipartite scenario proposed in reference [24]: (i) one black box that steers to two characterised devices and (ii) two black boxes that steer to one characterised device. inequality at its quantum maximum, then the following must hold: (i): (ii): G abcxyz P (b, c y, z, x, a) = λ (x,a) x, a (3.48) yzbc G abcxyz P (c z, x, y, a, b) = λ (x,y,a,b) x, y, a, b (3.49) zc and λ (x,a) and λ (x,y,a,b) are the largest eigenvalues of yzbc G abcxyzπ y b Πz c and zc G abcxyzπ z c respectively. We study the correlation that violates maximally the Mermin inequality [65] A 0 B 0 C 0 A 0 B 1 C 1 A 1 B 0 C 1 A 1 B 1 C 0 2 (3.50) 66

81 3.2. OW CRITERION IS AN ARTEFACT OF NO-SIGNALLING When rewritten to the form of a Bell expression, we get: B M (P ) := G abcxyz P (a, b, c x, y, z) 3 (3.51) abcxyz where G abcxyz = δ a b c,x y z δ xyz,000 δ xyz,011 δ xyz,101 δ xyz,110 (3.52) The maximal quantum bound of G M (P ) is given by 4 and it self-tests [72] the measured quantum to be the GHZ state GHZ = (3.53) and the measurements to be A 0 = B 0 = C 0 = σ z, (3.54) A 1 = B 1 = C 1 = σ y. (3.55) Using the above quantum state and measurements, one can easily check that: G abcxyz P (b, c y, z, x, a) = λ M(x,a) = 2 x, a (3.56) yzbc G abcxyz P (c z, x, y, a, b) = λ M(x,y,a,b) = 1 x, y, a, b (3.57) zc Thus, this concludes that Mermin inequality fulfils the OW criterion for both types of steering Further discussion In this chapter, we have shown that OW criterion holds only with Bell expression of a certain forms. By performing transformation that preserves equivalence under no-signalling principle, one may even change the OW character. On one hand, we have many counterexamples to the OW criterion. On the 67

82 CHAPTER 3. NONLOCALITY AND UNCERTAINTY RELATION other hand, all known counterexamples could be circumvented by performing a transformation on its Bell expression, which does not change anything under the no-signalling principle. This result heralds even more questions: can we always find a Bell expression that adhere to the OW criterion for every point on the quantum boundary? If so, what does this criterion even mean? Nevertheless, these questions are non-trivial. For one, the characterisation of the boundary of the set of quantum correlations still remains a mystery. 68

83 Chapter 4 Geometry of the Quantum Set 4.1 Background The desire for visualisation of the set of quantum correlations give rise to plots like Fig. 4.2 where the quantum set looks like a circle. While the plot is valid, it does not capture the rich geometric features of the quantum set. For instance, all boundary points on a circular-shape convex set are extremal and exposed. This mental imagery may misled one to believe that every maximal Bell violation is achieved by a unique point. The lack of proficiency in convex geometry also makes it difficult to reconcile with results like [57], where the authors proposed a sequence of Bell inequalities that captures the nonlocality of Hardy s paradox only when the terms in the Bell expression diverges. Later in this chapter, we will show that the unique point that maximises Hardy s paradox is not exposed which contributes to the result found in [57]. In this chapter, we aim to bring forth the different types of boundary points of the quantum set with explicit examples and its plots, in hope to shed some light on the complexity of the geometry of the quantum set. In order to do so, we will first introduce the notions that are used in the field of convex geometry. Then, we will recap on the definitions of the sets of correlations of interest (local polytope, no-signalling polytope and the quantum set) and furnish 69

84 CHAPTER 4. GEOMETRY OF THE QUANTUM SET new notations for convenience of the discussion here. Finally, we will give explicit examples of each type of boundary points (exposed, extremal but non-exposed and non-extremal boundary) that are present in the quantum set Convex set and its geometry In this subsection, we will introduce standard notions and definitions used in convex geometry. For more details, the reader can refer to Chapters 1 and 8 of reference [89]. Let A be a convex subset of R d and, moreover, suppose that A is compact (i.e. closed and bounded). For an arbitrary vector g R d let c( g) := max g u u A and note that the hyperplane { u R d : u g = c( g)} is a supporting hyperplane, i.e. it has a non-empty intersection with A and it divides the space into two halfspaces such that A is fully contained in one of them. The vector g represents a linear functional acting on R d. It is well known that every convex set can be described as an intersection of half-spaces (possibly infinite of them). Supporting hyperplanes help us to understand the boundary of the convex set. For an arbitrary functional g the set of points which maximise g, F( g) := { u A : g u = c( g)} is called an exposed face of A and since A is compact, the face is always non-empty. An exposed face is called proper if F( g) A. A point u A is called a boundary point if it belongs to some proper exposed face and we denote the set of boundary points by A bnd. The set of interior points of A is simply the complement of A bnd (in A). Some boundary points have the property that they cannot be written as a nontrivial convex combination of other points in the set. Such points are called ex- 70

85 4.1. BACKGROUND Non-extremal boundary point Exposed point Exposed point Convex Set Interior point Non-exposed extremal point Fig. 4.1: Different types of points of a compact convex set. tremal and we denote the set of extremal points by A ext. The Krein-Milman theorem states that any convex compact set (in a finite-dimensional vector space) is equal to the convex hull of its extremal points A = Conv(A ext ). Therefore, when maximising a linear functional over the set, it suffices to perform the optimisation over its extremal points. In other words, for all g we have max g u = max g u. g A g A ext Knowing the extremal points of A is also sufficient to determine its faces. Since a face is a convex compact set, it is equal to the convex hull of its extremal points and the extremal points of the face must also be extremal points of A. For exposed faces we have F( g) = Conv ( { u A ext : g u = c( g)} ). Among extremal points there are points which can be identified as unique max- 71

86 CHAPTER 4. GEOMETRY OF THE QUANTUM SET imisers of some linear functional. We say that u is exposed if there exists a linear functional g such that F( g) = { u} and we denote the set of exposed points by A exp. From the definitions alone, we immediately establish the inclusions A exp A ext A bnd A and it is well known that all of them are in general strict. However, it is worth pointing out that by Straszewicz s theorem in a finite-dimensional vector space the set of exposed points is dense in the set of extremal points [14]. In other words extremal but non-exposed points should be regarded as exceptional. For a polytope the set of extremal and exposed points coincide, as they are simply the vertices of the polytope The set of correlations In this chapter, we will focus on the [(2, 2), (2, 2)] Bell scenario and then the correlations can be assembled into a real vector P := ( P (a, b x, y) ) R 16, which we will refer to as the behaviour 1, probability point or simply a point. These nomenclatures are much more relevant in this context as each set of correlations corresponds to a point in the correlation space. Also, it is clear that all conditional probability distributions must be non-negative P (a, b x, y) 0 a, b, x, y (4.1) and normalised P (a, b x, y) = 1 x, y. (4.2) ab 1 Tsirelson first used the term behaviour [92] to describe a family of probability distributions indexed by tuples of setting values. The term has become widely adopted, e.g. in Refs. [69, 22]. The terms box [12, 13] and probability model [3] are also commonly used. 72

87 4.1. BACKGROUND The no-signalling set Following from the definition of no-signalling constraints in equations (1.1) and (1.2), a probability point belongs to the no-signalling set if it satisfies a, x, y, y P (a, b x, y) = b b b, x, x, y P (a, b x, y) = a a P (a, b x, y ) and P (a, b x, y). (4.3) Recall that the term no-signalling [12] refers to the fact that the choice of measurement settings of one party does not affect the outcome distribution of the other party. We denote the set of all no-signalling behaviours by N S and since it is characterised by a finite number of linear inequalities and equalities, namely (4.1), (4.2) and (4.3), the no-signalling set is a polytope known as the no-signalling polytope. The quantum set The quantum set Q is the set of correlations which can be achieved by performing local measurements on quantum systems. Following the standard tensor-product paradigm each party is assigned a Hilbert space H of finite dimension d := dim(h) <. A valid quantum state corresponds to a d 2 d 2 matrix which is positive semidefinite and of unit trace. A local measurement with outcomes is a decomposition of the d-dimensional identity into positive semidefinite operators, i.e. {Π Ax a } a=1 such that Π Ax a 0 for all x, a and a=1 where 1 d denotes the d-dimensional identity matrix. Π Ax a = 1 d, (4.4) We define Q finite to be the set of behaviours which can be generated when local Hilbert spaces are finite-dimensional, i.e. P Qfinite if there exists a finitedimensional quantum state ρ and measurements {Π Ax a }, {Π By b } such that P (ab xy) = tr [ (Π Ax a 73 Π By b )ρ ] (4.5)

88 CHAPTER 4. GEOMETRY OF THE QUANTUM SET for all a, b, x, y. To make the underlying mathematics neater, we define the quantum set Q as the closure of Q finite, i.e. we explicitly include all the limit points, which makes the quantum set Q compact. The fundamental result that Q Q finite for some finite Bell scenarios was only recently established by [90] (see also the recent work of Dykema et al. [39]). The local set Recall from equation (1.4), we call a probability point deterministic if the output of each party is a (deterministic) function of their input and we denote the set of deterministic points by L det. The local set is defined as the convex hull of the deterministic points L := Conv [L det ]. Since L det is a finite set, the local set L is a polytope, which is known as the local polytope. 4.2 Boundary points of the quantum set This section is based on the publication titled Geometry of the quantum set of correlations co-authored with Jędrzej Kaniewski, Elie Wolfe, Tamás Vértesi, Xingyao Wu, Yu Cai, Yeong-Cherng Liang and Valerio Scarani. In this section, we will focus on the simplest non-trivial Bell scenario, i.e. the [(2, 2), (2, 2)] Bell scenario. It is well known [42] that in this scenario the local set is fully described by the positivity inequalities (4.1), no-signalling constraints (4.3) and 8 additional inequalities, which are all equivalent (up to permutations of inputs and outputs) to the CHSH inequality [29]. The existence of a single type of (facet) Bell inequalities and the fact that any no-signalling probability point P N S can violate at most one of these inequalities [54] means that we can interpret the CHSH violation as a measure of distance from the local set. More specifically, Bierhorst 74

89 4.2. BOUNDARY POINTS OF THE QUANTUM SET showed that the total variation distance from the local set and the local content [84] can be written as linear functions of the violation [18]. Let us start by introducing convenient notation for the [(2, 2), (2, 2)] scenario. Correlations in the [(2, 2), (2, 2)] scenario are described by vectors P R 16, but due to the no-signalling constraints these vectors span only an 8-dimensional subspace and it is convenient to use a representation which takes advantage of this dimension reduction. Following the convention that a, b {0, 1}, we define the local marginals as A x := P (a = 0 x) P (a = 1 x), B y := P (b = 0 y) P (b = 1 y) and the correlators as A x B y := P (a = b xy) P (a b xy). The inverse relation is given by P (a, b x, y) = 1 4( 1 + ( 1) a A x + ( 1) b B y + ( 1) a+b A x B y ). (4.6) While this transformation is valid for any no-signalling point, the notation is inspired by quantum mechanics, since for a quantum behaviour the local marginals and correlators are simply expectation values ( X = tr(xρ)) of the local observables A x = Π Ax 0 Π Ax 1, B y = Π By 0 Π By 1 75

90 CHAPTER 4. GEOMETRY OF THE QUANTUM SET and their products. The expectation values are conveniently represented in a table B 0 B 1 P = A 0 A 0 B 0 A 0 B 1 A 1 A 1 B 0 A 1 B 1 (4.7) and it is natural to use the same representation when writing down Bell expressions. It is worth pointing out that the coordinate transformation that takes us from the conditional probabilities of events P (a, b x, y) to the local marginals ( A x, B y ) and correlators ( A x B y ) is a linear transformation, but it is not isometric. In other words, the transformation does not change any qualitative features of the set, e.g. whether a point is extremal or exposed, but it might affect measures of distance or volume. For a given correlation set S = L, Q, N S we denote the maximum value of the Bell expression B by β S ( B) := max P S B P. Note that since all these sets are compact, the maximum is always achieved. To simplify the notation we will simply write β S whenever the Bell expression is clear from the context. Bell expressions are useful for studying the three correlation sets, but they suffer from the problem of non-uniqueness, i.e. the same function can be written in multiple ways which are not always easily recognised as equivalent (see, however, reference [82]). To overcome this obstacle instead of studying the inequalities we study the (exposed) faces they give rise to. For a correlation set S = L, Q, N S every Bell expression B identifies a face F S ( B) := { P S : B P = β S }. All the sets considered here are compact, so the face is always non-empty, i.e. it 76

91 4.2. BOUNDARY POINTS OF THE QUANTUM SET contains at least one point. Since the local set L and the no-signalling set N S are polytopes, all their faces are also guaranteed to be polytopes, while for the quantum set Q this is not necessarily the case. The dimension of the face F S ( B) is simply the dimension of the affine subspace spanned by the points in F S ( B). While the dimensions of local and no-signalling faces are easy to compute (we simply find which vertices saturate the maximal value and then check how many of them are affinely independent), there is no generic way of computing the dimension of a quantum face. However, if the quantum value coincides with either the local or the no-signalling value, an appropriate bound follows directly from the well-known set inclusion relation, L Q N S. β Q ( B) = β L ( B) = dim ( F Q ( B) ) dim ( F L ( B) ) (4.8) and β Q ( B) = β N S ( B) = dim ( F Q ( B) ) dim ( F N S ( B) ). (4.9) A flat boundary region is an exposed face which contains more than a single point, but is strictly smaller than the entire set. 2 In the remainder of the section, we will look at various quantum boundary points in the [(2, 2), (2, 2)] scenario. As we will be dealing with many Bell expressions in this section, they will be enumerated for consistency. The plots of the quantum set presented in this section are obtained by running two types of numerical optimisation: (1) the NPA relaxation [69] provides the upper bound of the quantum boundary, (2) heuristic optimisation of the axes of plots over quantum states and measurements yield a lower bound of the quantum boundary. When there are gaps between the results of (1) and (2), the quantum boundary is obtained. Also, when the result of (1) yields a flat boundary where the endpoints are known to be tight, the quantum boundary is also obtained (the quantum set is convex). 2 One should take care not to confuse the face F S ( B) with the hyperplane B P = β S : the face F S ( B) is the intersection of S with the hyperplane. 77

92 CHAPTER 4. GEOMETRY OF THE QUANTUM SET Exposed points The most well-known boundary points on the quantum set are the exposed points and there are no lack of examples of them. For completeness, we present the CHSH example here. The CHSH point The famous CHSH Bell expression [29], in the new notation, is given by B 1 := L, B1 P 2 2 Q. (4.10) N S This inequality is known to have a unique quantum maximiser [64, 22] 0 0 P CHSH := , (4.11) which implies that P CHSH is an exposed point of the quantum set. In Fig. 4.2 we show P CHSH in the 2-dimensional slice spanned by 4 variants of the Popescu-Rohrlich 78

93 4.2. BOUNDARY POINTS OF THE QUANTUM SET (PR) box [12, 13, 10, 22, 79] P PR := 0 1 1, PPR, 2 := 0 1 1, (4.12) P PR, 3 := 0 1 1, PPR, 4 := The central point of this plot corresponds to the uniformly random distribution, Fig. 4.2: A 2-dimensional slice in which the quantum set exhibits no flat boundaries (first presented as Fig. 1 in reference [19]). Points on this slice can also be conveniently parametrised by two different versions of the CHSH Bell expressions as in Fig. 3 of reference [26]. 79

94 CHAPTER 4. GEOMETRY OF THE QUANTUM SET which can be written as a uniform mixture of the 4 PR boxes, i.e. 0 0 P 0 := (4.13) Non-extremal boundary regions Recall that a non-extremal boundary or flat boundary region is an exposed face which contains more than a single point, but is strictly smaller than the entire set. Here, we will present two explicit examples of such regions. Region between CHSH and deterministic point Now we consider the Bell expression B 2, which is given by B 2 := L, B2 P 4 Q, (4.14) N S where the local and no-signalling bounds have been computed by enumerating the vertices of the polytopes, while the quantum bound has been computed using the analytic technique of Wolfe and Yelin [96]. The quantum bound is saturated by P CHSH but also by the deterministic point 1 1 P d,1 := (4.15) This implies that the resulting quantum face is at least 1-dimensional and we conjecture that this lower bound is actually tight, i.e. that the quantum face is a line. 80

95 4.2. BOUNDARY POINTS OF THE QUANTUM SET Fig. 4.3 shows this quantum face in the slice containing P CHSH, P 0 and P d,1 (the same feature was presented in Fig. 3(c) of reference [38]) ( + ) + + Fig. 4.3: A slice containing P CHSH, P0 and P d,1 is singled out by the following six equations: A 0 = A 1 = B 0 = B 1, A 0 B 0 = A 0 B 1 = A 1 B 0 and A 0 + A 1 + B 0 + B 1 = 2( A 0 B 0 + A 1 B 1 ). The point PL,1 is given by A x = B y = A x B y = 1. Apart from the flat quantum face F 3 Q( B 2 ) that connects P CHSH and P d,1, our numerical results suggest a few other flat regions on the boundary of (this slice of) the quantum set. The above quantum face is not the only flat face containing P CHSH. To see this we swap the outcomes of all the measurements: this results in flipping the horizontal axis of Fig. 4.3 while leaving the vertical axis unchanged. This relabelling transforms 81

96 CHAPTER 4. GEOMETRY OF THE QUANTUM SET ( + )+ + Fig. 4.4: Projection illustrating the flat boundary identified by B 1. While every point in a slice plot corresponds to precisely one behaviour, a point in a projection plot may simultaneously represent multiple behaviors. In this projection the behaviours P CHSH and P d,1 are the only behaviours that lie on the points (0, 2 2) and (4 2 2, 2), respectively. We see that all three correlation sets are symmetric with respect to the reflection about the x = 0 line. This symmetry arises from the relabelling of the outcomes of all measurements, which flips the marginals, but leaves the correlators unchanged. the Bell expression B 2 to B 2 := L, B2 P 4 Q, N S 82