Thesis
Reference
Characterization of correlations in quantum networks
ROSSET, Denis
Abstract
Les derniers développements en physique quantique ont permis l'établissement de réseaux quantiques sécurisés composés de plusieurs sources et observateurs. D'étonnantes corrélations sont observées dans ces réseaux; jusqu'à récemment, la description théorique qui en était faite ne tenait pas compte de la topologie particulière des réseaux considérés. Le sujet principal de cette thèse est l'étude de ces corrélations selon des topologies variées. La thèse comporte également d'autres pistes de recherche, telles que le classement des inégalités de Bell et la caractérisation des systèmes quantiques.
ROSSET, Denis. Characterization of correlations in quantum networks. Thèse de doctorat : Univ. Genève, 2015, no. Sc. 4813
URN : urn:nbn:ch:unige-774011
DOI : 10.13097/archive-ouverte/unige:77401
Available at:
http://archive-ouverte.unige.ch/unige:77401
Disclaimer: layout of this document may differ from the published version.
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Université de Genève Faculté des sciences
Groupe de physique Appliquée Professeur N. Gisin
Characterization of Correlations in Quantum Networks
THÈSE
présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences,
mention physique
par
Denis ROSSET de Genève (GE)
Thèse No 4813
Genève
Atelier Repromail, Université de Genève 2015
Literally: things will be gathered together according to category (Yi Jing).
Résumé
Le sujet de cette thèse est la caractérisation des corrélations obtenues dans des réseaux composés de sources quantiques et d’appareils de mesure séparés spa- tiallement. Plusieurs types de réseaux sont considérés, correspondant à différentes topologies de distribution, ainsi qu’à plusieurs familles d’hypothèses à propos de la structure causale régissant les processus impliqués. Dans ces réseaux, nous inves- tiguons les ensembles de corrélations observables lorsque les sources et les appareils respectent des principes généraux contraignant leur mode de fonctionnement, sans pour autant considérer dans le détail le fonctionnement interne des appareils.
Un principe majeur dans l’étude de l’information quantique est le principe de localité, qui prescrit que les objets physiques sont influencés par leur environnement proche uniquement. Le principe de localité est souvent étudié sous la forme du théorème de Bell, qui construit des tests statistiques particuliers connus sous le nom d’inégalités de Bell. Ensuite, les corrélations obtenues dans des réseaux peuvent être examinées avec l’aide de ces inégalités : ce faisant, il a été démontré que la mécanique quantique prédit des corrélations qui violent ces inégalités de Bell, falsifiant en conséquence le principe de localité.
Bien que notre recherche ait été conduite au niveau des fondements de l’infor- mation quantique, nous avons été amenés à construire des méthodes qui peuvent être implémentées expérimentalement. Nous avons étudié des généralisations du théorème de Bell dans plusieurs directions. Nous avons généralisé la localité de Bell à des réseaux impliquant plusieurs sources indépendantes, un phénomène qui n’avait pas été complètement capturé par la formulation du principe de localité dans le théorème de Bell. Ces contraintes additionelles ont mené à des tests plus robustes de la non-localité en réseau. Nous avons également effectué des premiers pas vers une formulation d’inégalité générales dans des réseaux à topologie quelconque, un développement requis par les avancées expérimentales réalisées pour la distribution d’intrication à distance, en utilisant des répéteurs quantiques. En collaboration avec Gilles Pütz, nous avons étudié la relaxation d’une hypothèse clé du théorème de Bell correspondant à l’indépendence du choix des mesures.
Dans une deuxième partie de la thèse, nous avons étudié une propriété fondamen- tale des états quantiques, l’intrication, qui est requise pour que ces états aient une utilité dans les réseaux quantiques. Nous avons d’abord montré que les outils stan- dards utilisés pour la caractérisation de l’intrication étaient susceptibles de donner des résultats faux-positifs lorsque des erreurs systématiques sont présentes. Nous avons proposé plusieurs solutions à ce problème : tout d’abord, nous avons généralisé le théorème de Bell en employant des entrées quantiques et non classiques pour spécifier les mesures à effectuer. Dans ce contexte, nous avons montré que tous les états intriqués peuvent être certifiés, en étant robuste face aux erreurs systématiques, face aux pertes et aux inefficacités de détection, face au cross-talk entre appareils (pour autant que ce cross-talk soit modélisé par de la communication classique).
5
Dans les réseaux impliquant une source distribuée à des multiples observateurs, nous avons étudié deux phénomènes. Le premier concerne l’activation de non-localité présente dans un état quantique, en utilisant un autre état comme catalyste ; en collaboration avec Yeong-Cherng Liang, nous avons montré que tout état intriqué contient de la non-localité cachée, en présentant différents exemples. Deuxièmement, en collaboration avec nos co-auteurs, nous avons étudié la structure de l’intrication dans des états multipartites et avons construit une famille d’inégalités de Bell pour certifier l’étendue de l’intrication dans un état quantique.
Finalement, nous avons investigué le rôle des symétries dans les scénarios de corrélations. Bien que l’étude des symétries soit un domaine central en physique théorique, son rôle dans la non-localité n’avait pas été exploré en détail. Nous avons proposé une définition formelle des symétries des scénarios de corrélation, et étudié le lien entre les symétries d’une inégalité de Bell et ses propriétés structurelles.
Nous avons également démontré d’importants gains de vitesse dans l’emploi de la hiérarchie de relaxation semi-définie pour le calcul de bornes quantiques. Les symétries, entre autres éléments, introduisent des dégénérescences dans la descrip- tion des inégalités de Bell. Nous avons construit un procédé pour classer les inégalités connues et les compiler dans une base de données.
6 Résumé
Abstract
The subject of this thesis is the characterization of correlations obtained in net- works composed of spatially separated sources of quantum states and measurement devices. Several kinds of networks are considered, corresponding to different topolo- gies in the distribution of states to devices, and to different assumptions about the causal structure constraining the physical processes involved. In those networks, we investigate the sets of correlations obtained when the sources and devices respect general principles ruling their operation, without considering the inner workings of the subsystems in detail.
A major principle studied in quantum information is the principle of locality, pre- scribing that devices are directly influenced only by their immediate surroundings.
The locality principle is often studied under its formulation in Bell’s theorem, leading to statistical tests known under the name of Bell inequalities. In turn, correlations in networks are tested using Bell inequalities, and it was demonstrated that quantum correlations can violate Bell inequalities, thus falsifying the principle of locality.
While our research is done at the fundamental level, it led us in many cases to discover practical methods that can be applied experimentally. We generalized Bell locality to networks including multiple independent sources, a feature that was not captured completely by the formulation of the locality principle in Bell’s theorem. These additional constraints led to stronger tests of quantum nonlocality in networks. We also showed steps towards a formulation of Bell-like inequalities in quantum networks with any topology, a construction required by the latest devel- opments in entanglement distribution networks using quantum repeaters. In collabo- ration with Gilles Pütz, we studied the relaxation of the independence of the choice of measurement settings in Bell tests.
In the second part of this thesis, we studied the property required for a quantum state to be useful in a quantum network, namely entanglement. As quantum states are implemented experimentally in many different physical systems, robust methods to characterize entanglement are vital to assess the merits of these implementations.
As we show however, standard tools used for entanglement certification fail in the presence of systematic errors in the measurements. We then proposed several direc- tions to correct this problem. One direction involves another generalization of Bell’s theorem, in which the measurements performed by the devices are prescribed using well-characterized quantum input states. In this context, all entangled states can be certified by a method robust against systematic errors in the measurements, losses, detection inefficiencies and (classical) crosstalk between devices.
7
In networks involving a single source distributing states to multiple parties, we studied two phenomena. The first involves the activation of nonlocality hidden in a quantum state, using another state that acts as a catalyst; in collaboration with Yeong-Cherng Liang, we showed that all entangled states contain such hidden nonlocality, and constructed several examples. The second concerns the structure of entanglement in multipartite state; also in collaboration with other co-authors, we exhibited a Bell inequality that certifies the extent of multipartite entanglement.
Finally, we investigated the role of symmetries in correlation scenarios. While symmetry is a central principle in physics, its role in nonlocality has not been explored in depth. We proposed a formal definition of symmetries in correlation scenarios and studied the link between symmetries and other properties of Bell inequalities. We also demonstrated huge computational gains in a class of numer- ical methods widely used in quantum information, namely semidefinite relaxations.
Symmetries, among other elements, also induce degeneracies in the description of Bell inequalities. We devised a scheme to classify the currently known inequalities and enter them into an online database.
8 Abstract
Table of contents
Résumé . . . 5
Abstract . . . 7
1 Introduction . . . 13
I n-locality
. . . 212 Introduction to independence condtions . . . 23
2.1 Towards n-locality . . . 24
2.2 Weakened measurement independence . . . 25
2.3 Conclusion . . . 28
3 Bilocality and entanglement swapping . . . 29
3.1 A bilocal scenario for entanglement swapping . . . 30
3.2 Bilocal inequalities and quantum violations . . . 32
3.3 Mathematical characterization of bilocal sets . . . 34
3.4 Outlook . . . 35
4 Simulations of bilocal correlations . . . 37
4.1 Bilocal simulation of entanglement swapping . . . 38
4.2 Outlook . . . 39
5 Inequalities for n-local networks . . . 41
5.1 Extensions of n-local inequalities . . . 42
5.2 n-locality in a chain network . . . 43
5.3 Construction walkthrough . . . 44
5.4 Outlook . . . 46
6 Towards general n-local scenarios . . . 47
6.1 The triangle: simplest cyclic n-local scenario . . . 47
6.2 General n-local scenarios . . . 50
6.3 Outlook . . . 50
II Imperfect devices
. . . 537 Effects of imperfect measurement settings . . . 55 9
7.1 Systematic errors and entanglement witnesses . . . 56
7.2 Systematic errors and quantum tomography . . . 58
7.3 Outlook . . . 59
8 MDIEWs - entanglement detection without trusted measurement devices . . . 61
8.1 Semiquantum scenarios . . . 62
8.1.1 Extracting entanglement out of a semiquantum scenario . . . 63
8.1.2 Ideal semiquantum scenarios . . . 64
8.2 Measurement-device-independent entanglement witnesses . . . 65
8.2.1 Examples of MDIEWs . . . 66
8.2.2 Non-tomographically complete sets of input states . . . 68
8.3 MDIEWs for multipartite entanglement . . . 68
8.4 Outlook . . . 69
9 MDIEW construction and entanglement quantification . . . 71
9.1 MDIEW construction from experimental data . . . 71
9.1.1 When non-detections are not observed . . . 72
9.2 Quantitative entanglement estimation . . . 73
9.2.1 Extractable entanglement from local filtering . . . 73
9.2.2 Estimation of extractable negativity . . . 73
9.3 Outlook . . . 75
III Structure of Bell inequalities
. . . 7710 Symmetries of Bell inequalities . . . 79
10.1 Relabelings . . . 79
10.1.1 Types of relabelings . . . 80
10.1.2 Groups of relabelings . . . 80
10.1.3 Actions on probability distributions, strategies . . . 81
10.2 Symmetry groups of Bell inequalities . . . 81
10.2.1 Recognizing liftings by the symmetry group . . . 82
10.3 Symmetric SDP relaxations . . . 82
10.3.1 Quantum bound of CHSH: a linear program . . . 83
10.3.2 NPA bounds on I3322in extended precision . . . 84
10.3.3 Verified SDP bounds and symmetrization . . . 85
10.4 Inequalities invariant under permutation of parties . . . 86
10.4.1 Examples . . . 87
10.5 Outlook . . . 88
11 Classification of Bell inequalities . . . 89
11.1 Steps in the decomposition . . . 91
11.2 Outlook . . . 92
IV Other results
. . . 9310 Table of contents
12 Multipartite hidden nonlocality . . . 95
12.1 Definitions . . . 96
12.2 Main result . . . 96
12.3 Examples . . . 97
12.4 Outlook . . . 98
13 Bell inequalities for entanglement depth . . . 101
13.1 Outline of the result . . . 102
13.2 Outlook . . . 103
14 Conclusion . . . 105
Remerciements . . . 107
V Appendices
. . . 109Appendix A Bilocal inequality for inefficient detectors . . . 111
Appendix B Elimination of quantifiers . . . 115
Appendix C Transforms of n-local models . . . 117
C.1 Definitions . . . 117
C.1.1 Scenarios . . . 117
C.1.2 Correlations . . . 118
C.1.3 Models . . . 118
C.2 Internal transforms of n-local models . . . 119
C.2.1 (PRC) Performing the processing in the LHV . . . 119
C.2.2 (PUL) Simplification of partially-used finite LHV . . . 120
C.2.3 (DET) Making processing functions deterministic . . . 120
C.2.4 (FIN) Making a LHV finite . . . 120
C.3 Transforms that relate different scenarios . . . 122
C.3.1 (FLI) Flattening of a party input . . . 122
C.3.2 (RML) Removal of a finite LHV . . . 123
Appendix D Constructions of n-local models . . . 125
D.1 2-local models . . . 125
D.1.1 2-local models for binary outputs . . . 126
D.2 Generic 3-local models for the triangle . . . 127
D.2.1 First model by general construction . . . 127
D.2.2 Second model by reduction to bilocality . . . 128
Appendix E Semialgebraic sets for n-local correlations . . . 131
E.1 Semialgebraic sets of finite n-local correlations . . . 131
E.2 Tools and algorithms for semialgebraic sets . . . 132
E.3 All n-local scenarios have finite models . . . 133
Table of contents 11
Appendix F Characterization of quantum memories using trusted state
preparation devices . . . 135
F.1 Introduction . . . 135
F.2 Setup . . . 135
F.3 Witness construction . . . 136
Appendix G Binary forms of Bell inequalities . . . 139
G.1 Finding binary forms of inequalities . . . 140
G.2 Result for Sliwa’s 46 families . . . 141
Bibliography . . . 147
VI Published papers
. . . 15712 Table of contents
Chapter 1 Introduction
Most physical theories are concerned with the modelization of phenomena by approximating physical proceses using their key mechanisms. Quantum mechanics, constructed to describe the outcomes of measurements performed on atomic-scale systems, was met by wide skepticism; not because of failures of its predictive power, but because the theory seemingly transgressed many principles and intutions we take for granted about the world.
One of these principles is the principle of locality: objects are directly influenced only by their immediate surroundings. Cutting a long story short, the debate of the locality of physical theories dates back to the theory of gravitation. As formulated by Newton in the 17thcentury, it involved celestial bodies acting on each other from a distance, without the “mediation of anything else” [11], a description that bothered Newton himself. To remedy this problem, Faraday introduced his Nahwirkung- sprinzip (near-effect-principle) around 1852, to explain the attraction/repulsion of objects by the propagation through space of an interaction mediator. The idea was taken on with success by Maxwell and led to the theory of classical electro- dynamics a decade later. At the beginning of the 20th, Maxwell’s equations inspired Einstein to formulate the theory of special relativity, which stipulated that the speed of information and matter propagation is bounded by the speed of light, thus complementing nicely the Nahwirkungsprinzip/locality principle.
While classical electrodynamics is perfectly adequate to describe the behavior of light as we experience it in everyday life, it failed however to explain several effects observed from the measurements of individual photons (and other particles). This led to the formulation of the quantum theory in the first decades of the new century.
Quantum theory represented a radical departure from the principle of locality:
it models physical systems using a global representation incompatible with the prin- ciple of locality (respecting however special relativity). Moreover, quantum theory predicted that an action on one part of an experiment would have immediate con- sequences on distant subsystems, reviving the uncomfortableaction-a-distance trick which displeased Newton centuries before. The new theory was met with skepti- cism, including by Eistein, Podolsky and Rosen, who published in 1935 a paper [12]
proposing the following dilemna: either quantum mechanics is correct and we have 13
to reject the principle of locality, or quantum mechanics is only a step towards a saner theory that would restore the locality principle.
In 1964, John Bell proposed an experimental test of the locality principle, by investigating the correlations coming from spatially separated systems [13]. The test does not require any assumption about the underlying physical theory, nor about the workings of the experimental devices. Instead, John Bell proposed to use the principles of locality and of no-faster-than-light-communication to construct a model of the correlations that could be observed when the principles are obeyed:
his main theorem demonstrated that local correlations would satisfy statistical tests known today as Bell inequalities, which in turn could be violated by correlations predicted by quantum mechanics. By reproducing these quantum correlations in an experiment, the locality principle could be falsified. Such a demonstration would provide a definite answer to the EPR dilmena, as Bell’s derivation followed from the assumed structure of causes and effects in experimental setups, without assumptions about the underlying physical theory.
Since Bell’s proposal, researchers began to study the causal structures modeling several kinds of physical setups: instead of trying to approximate the inner work- ings of states and measurement devices involved in an experiment in a bottom-up approach, one instead starts by enumerating assumptions about the structure of causes and effects in the system, then studying the resulting constraints on observed correlations.
This approach led to major results in three areas. First, by abstracting the low- level details of physical systems, fundamental insights are gained about the nature of correlations [14, 15, 16]. Second, methods derived from Bell inequalities can be used to characterize quantum states while being robust against imperfections in the measurement process [17]. Finally, it was observed that violations of the locality principle could be taken as resources to be used in the implementation of information tasks, such as the distribution of secret keys [18] or the extraction of random bits in a secure way [19, 20].
We present in this thesis results on all these areas.
Fundamental notions: entanglement and nonlocality. — A key feature of quantum states is entanglement, present when a state composed of distinct parts cannot by described by a probabilistic mixture of separated parts. Entanglement is described in the context of quantum mechanics, assuming that prescribed quantum measurements can be performed on the state; its presence is often certified by the usage entanglement witnesses, making use of the data coming from precisely cali- brated quantum measurements.
At the level of correlations, a similar concept is callednonlocality; it is defined by treating measurement devices as black boxes, without assuming any specific physical theory, purely from the principles defining the causal structure of a scenario.
The relationship between entanglement and nonlocality is a persistent fun- damental question in quantum information. For example, nonlocality certifies indu- bitably the presence of entanglement without assumption about the measurement
14 Introduction
LASER RADIATION
choice of measurement
result
i) Experimental setup
ii) Causal model: Bell locality
iii) Quantum networks
Ex: SwissQuantum Ex: SwissQuantum
CERN
UNIGE HEPIA
iv) Causal models for quantum networks
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cFigure 1.1. In (i), we show the diagram of a photonic experiment composed of a source (laser + nonlinear optical element) and two measurement devices. The corresponding causal model (ii) can be used to test Bell’s notion of locality. In (iii), we show as an example the application of quantum physics to communication networks with a more complex graph. These correlations can currently be described only partially (iv).
performed: a feature at the heart of all device-independent protocols [17].
Yet, some entangled states never violate Bell’s locality, because their correlations can always be simulated using a model respecting the principle of locality. The device-independent certification of those states is thus impossible in this context.
Locality applied to complex networks. — The principle of locality was studied by Bell in a bipartite system. Extensions were made to multipartite sys- tems, most often using a single source to send a state to all observers in the system.
As the applications of quantum correlations grow, more complex networks are being implemented, for example to distribute states using entanglement swapping, with repeaters employing multiple independent sources of entanglement [21]. Other tested quantum networks have loops in the links between observers (see Figure 1.1). The principle of locality was not used to its fullest extent to describe the correlations in such networks. In [22], Branciard et al. analyzed the consequences of a strict characterization of the locality principle, by assuming that independent sources are uncorrelated. This led to the notion of n-locality, in scenarios where n inde- pendent sources are present.
In Chapter2, we start by reminding that an independence assumption is already present in Bell’s theorem: measurement independence. We show that n-locality fol- lows the spirit of the locality principle, and discuss the role of human observers in Bell tests and their replacement by random number generators. In collaboration with our colleague Gilles Pütz, we study the relaxation of the measurement independence assumption.
Starting from Chapter3, we analyzen-locality in depth, starting from2-locality,
Introduction 15
or bilocality. It is shown in Chapter 3 that stronger quantum violations can be obtained in bilocality, compared to Bell locality, using robustness against noise and detection inefficiencies as a benchmark. In Chapter 4, we quantify the strength of the nonlocal correlations obtained by the number of bits of classical communication needed for their simulation. More generaln-local networks are studied in Chapter5, by the iterative extension of inequalities valid for simpler scenarios. The approach of Chapter 5, however, only applies to networks with a tree-like structure and fails when the graph of observers and sources contain cycles, as shown in Figure 1.1iv.
The additional difficulties encountered are discussed in Chapter6, where it is shown that all n-local scenarios can be described using semialgebraic sets bounded by polynomial inequalities.
Imperfect measurements and entanglement certification. — The entan- glement witnesses described previously are widely used to certify the presence of entanglement in quantum states. Entanglement is a key feature of many quantum information protocols and at the basis of the additional power offered by quantum computing; as quantum states have implementations in a variety of different physical systems, there is intense competition to certify the presence of entanglement in multipartite states of increasing size. However, standard entanglement witnesses require precisely calibrated measurements. We show in Chapter7that the presence of systematic errors in the measurements can lead to the incorrect certification of entanglement in separable states, and describe a procedure to correct the bounds of witnesses to increase their robustness.
Bell scenarios with quantum inputs. — In Bell locality, observers employ devices that perform measurements according to the instructions they receive. These instructions are provided using inputs composed of classical information (e.g. bits).
Francesco Buscemi introduced semiquantum scenarios in 2012 [23], modifying Bell scenarios by allowing observers to use quantum states as measurement settings. In the new setups,all entangled states can exhibit nonlocal-like correlations. However, Buscemi’s proof is not constructive, and cannot be used in practical applications.
In Chapter 8, we present a constructive proof to Buscemi’s result. We define measurement-device independent entanglement witnesses (MDIEWs), standing half- way between standard entanglement witnesses and device-independent approaches.
Contrary to device independent entanglement witnesses, MDIEWs can detect all entangled states. Contrary to standard entanglement witnesses, they are robust against imperfections in the measurement devices, including systematic errors and detection inefficiencies, relying instead on the calibrated local preparation of single- particle states. Suprisingly, MDIEWs are also robust when crosstalk between devices is present, if this crosstalk can be described by classical information sent among them.
In their first formulation, MDIEWs certify the presence of entanglement but do not offer a way to quantify it. In Chapter 9, we show how to construct MDIEWs purely from experimental data to provide lower bounds on the entanglement that can be extracted from the state under study.
16 Introduction
Symmetries and classification of Bell inequalities. — Coming back to the local scenarios described by Bell, we study the structure of Bell inequalities. We consider the symmetries involved in Bell scenarios in Chapter 10. We display the symmetry groups associated with different inequalities, and find various relations between the group and the structure of an inequality. We also symmetrize the semidefinite relaxations used to compute the maximal quantum violations of Bell inequalities, dramatically lowering their CPU and memory requirements.
Insights gained from the symmetry structure are used in Chapter 11to provide the first classification of Bell inequalities, considering in addition other possible degeneracies present in their description. This classification was implemented in software, and is currently used to build a database of the inequalities published in the current literature (see http://faacets.com).
Multipartite nonlocality. — While some entangled states can never violate Bell locality by themselves, it is known that nonlocality can be hidden in a state, to be revealed when several of these states are used together. Masanes et al. showed that all bipartite entangled states could display such a phenomenon [24]. In collab- oration with Yeong-Cherng Liang, we extend this result to the multipartite case in Chapter 12along with a few examples.
We consider the structure of multipartite entanglement in Chapter 13, and, in collaboration with co-authors, construct a new family of Bell inequalities that can certify that at leastkparties are entangled together in a multipartite state, a feature termed entanglement depth.
Structure of the current thesis. — In this thesis, each Chapter provides the summary of an individual result and is self-contained, with the exception of Chap- ters 2to 6which should be read together, and similarly Chapters 8and 9. Most of our results have been published, with the corresponding papers numbered [1] to [10];
the reader should refer to them for more details. Technical results presented in this thesis are relegated to the Appendix.
The next two pages provide a graphical display of the scenarios considered in the thesis, and link the content of the papers with the corresponding Chapters.
Introduction 17
Quantum scenario
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Figure 1.2. Scenarios considered in the present thesis, with the corresponding chapters (white circles) and publications (black circles).
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Measurement-Device-Independent Entanglement Witnesses for All Entangled Quantum States
Branciard, Rosset, Liang,Gisin PRL 110, 060405 (2013)
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All entangled states display some hidden nonlocality
Liang, Masanes, Rosset PRA 86, 052115 (2012)
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Family of Bell-like Inequalities as Device-Independent Witnesses
for Entanglement Depth Liang, Rosset, Bancal, Pütz, Barnea, Gisin PRL 114, 190401 (2015)
① Introduction
② N-locality and measurement independence ❶❼
③ Bilocality ❶
④ Simulations of bilocal correlations ❶❽
⑤ Inequalities for n-local networks ❷
⑥ Towards general n-locals scenarios
⑦ Effects of imperfect measurement settings ❹
⑧ MDIEWs - entanglement detection without trusted detectors ❺❻
⑨ MDIEW construction from experimental data
⑩ Symmetries of Bell inequalities
⑪ Classification of Bell inequalities ❸
⑫ Multipartite hidden nonlocality ❾
⑬ Bell inequalities for entanglement depth ❿
Ⓐ Bilocal inequality for inefficient detectors
Ⓑ Elimination of quantifiers ❷
Ⓒ Transforms of n-local models
Ⓓ Construction of n-local models
Ⓔ Semialgebraic sets for n-local correlations
Ⓕ Characterization of quantum memories using trusted state preparation devices
Ⓖ Integer forms of Bell inequalities
Table of contents of this PhD thesis
Figure 1.3. Chapters are numbered 1-13, Appendices A-G in white circles. Content corresponding to published papers/drafts is indicated with black circles;
previously unpublished content written in italics. Papers at the core of the thesis are presented first.
Arbitrarily Small Amount of Measurement Independence Is Sufficient to Manifest
Quantum Nonlocality Pütz, Rosset, Barnea, Liang, Gisin
PRL 113, 190402 (2014) Bilocal versus nonbilocal correlations in
entanglement-swapping experiments
Branciard, Rosset, Gisin, Pironio PRA 85, 032119 (2012)
Classical Simulation of Entanglement Swapping with Bounded Communication Branciard, Brunner, Buhrman, Cleve, Gisin,
Portmann, Rosset, Szegedy PRL 109, 100401 (2012) Nonlinear Bell inequalities tailored
for quantum networks
Rosset, Branciard, Barnea, Pütz, Brunner, Gisin arXiv:1506.07380 (2015)
Imperfect measurement settings:
Implications for quantum state tomography and entanglement
witnesses Rosset, Ferretti-Schöbitz,
Bancal, Gisin, Liang PRA 86, 062325 (2012)
Entangled states cannot be classically simulated in generalized Bell experiments with quantum inputs Rosset, Branciard, Gisin, Liang
NJP 15, 053025 (2013) Classifying 50 years of Bell inequalities
Rosset, Bancal, Gisin JPA 47 424022 (2014)
Part I
n-locality
Chapter 2
Introduction to independence cond- tions
As discussed in the introduction, John Bell derived local models for correlations observed in scenarios obeying the locality principle. In turn, quantum mechanics predicts experimental observations that cannot be described by those models — such observations then falsify the principle of locality.
Several assumptions are made in the construction of local models. For example, it is assumed that the devices cannot communicate during the measurement process — indeed, nonlocal correlations could be simulated with the help of communication (see Figure 2.1c, and specific examples in Chapter 4). To disallow communication, the parties and their devices are separated in space, so that the selection of settings and the subsequent measurement can take place in a space-like separated manner [25], ensuring that no communication is possible due to the constraints on the speed of information as dictated by special relativity.
Another assumption is measurement independence, stipulating that the settings are chosen completely independently of any local hidden variables describing the system. This requirement is satisfied in principle by having human observers selecting measurement settings using their free will, as depicted in Figure 2.1a. However, this poses two challenges. First, human beings are notoriously poor sources of entropy (for an early review, see [26], or try http://www.loper-os.org/bad-at-entropy/manmach.html). Second, the same observers should select settings and perform measurements quickly enough to ensure space-like separation — in roughly a millisecond. Thus, the choice of settings is
a b
x y
a) Bell test (human observers)
a b
X Y
x y
b) Bell test (RNGs)
Alice device
Bob
device Alice
device
Bob device
a b
x y
c) Communication loophole
Figure 2.1. In a), Bell tests can be driven by human observers exercing their free will. In the variant b), measurement settings are instead replaced by random number generators, ensuring fast operation of the devices. Indeed, the devices have to be operated fast enough to close the locality loophole (c).
23
delegated in practice to fast, non-deterministic random number generators, con- structed to be independent of the other subsystems used in the experiment. Such an experiment is drawn in Figure 2.1b; then, the measurement independence con- dition no longer relies on human free will, but on the independence of the random number generators – as stated by Bell in 1977 [27, Chapter 12], these generators should be at least effectively free for the purpose at hand.
2.1 Towards n-locality
The original model by Bell thus assumes the existence of two co-existing, indepen- dent physical processes. The first concerns the selection of measurement settings, whether this choice is performed by human observers or by random number gen- erators. The second describes the state measured by the devices, without many restrictions: the devices could even measure the “state of the universe”, with the exception of systems involved in the settings selection.
As introduced by Branciard et al. in [22] and further elaborated in our joint work [1], this assumption can be formally defined. We first observe that experiments are described using sources and measurement devices, connected according to a specific network. In the spirit of the locality principle (objects interact only with their surroundings), we associate different local hidden variables to the sources, and assume that independent sources are described by uncorrelated local hidden vari- ables. This assumption is called n-locality, where n is the number of sources/local hidden variables in the model under consideration.
For example, a Bell test driven by random number generators satisfying measure- ment independence, as shown in Figure2.2a, is equivalent to a3-local scenario using three sources, drawn in Figure 2.2b. There, the value of the setting x is chosen by the source corresponding to λX, and sent to Xavier, acting as a clerk, and to Alice.
The 3-locality assumption, along with space-like separation, ensures measurement independence, and both models are equivalent, as developed in [1].
Note that n-locality has other uses than justifying measurement independence in Bell tests; it is particularly suited to the description of quantum networks, where several sources distribute quantum states to distant observers, as we will see in Chapters 3,4, 5 and 6.
a b
X Y
x y
a) Bell test (RNGs)
a
Alice
b
Bob
x
Xavier
y
Yolanda
b) Equivalent 3-local scenario
Alice device
Bob device
Figure 2.2. A Bell test using random number generators (a), and the equivalent 3-local scenario (b).
24 Introduction to independence condtions
2.2 Weakened measurement independence
We now turn to the effect of relaxing the measurement independence assumption.
When the measurement settings are completely influenced by the local hidden vari- ables, these variables can prescribe arbitrary values for all settings and outcomes at every step of the experiment; they are thus able to simulate any given probability distribution of settings/outcomes — including distributions that are seemingly non- Bell-local.
Now, suppose that the measurement independence assumption is only partially relaxed, and that partial measurement independence is available — for some quan- titative definition of measurement independence to be defined. Is it possible to relax the hypotheses of Bell’s theorem, and derive inequalities that can detect nonlocality when partial measurement independence is present?
Note:without any measurement independence whatsoever, the measurement settings are dictated by local hidden variables in a particular form of hard determinism. In this case, the study of quantum nonlocality is pointless [28]. While such a possibility can never be falsified, the subsequent question would be the meaning of scientific inquiry in a deterministic universe where the results of all experiments are prerecorded. Amusingly, the University of Geneva, where originated significant results about nonlocality, was founded in 1559 by a theologian with his own brand of determinism, Jean Calvin.
The question is not new. Hall [29] constructed a model of the singlet correla- tions using relaxed measurement independence, and computed relaxed bounds of the CHSH andImm22inequalities [30]. Barrett and Gisin [31] connected models involving relaxed measurement independence with models allowing classical communication or exploiting the detection loophole. Thinh et al. [32] defined the amount of measure- ment independence present using min-entropy sources, and derived relaxed bounds of the CHSH, CGLMP [33] and Mermin [34] inequalities.
A closely-related line of research is randomness amplification, a quantum pro- tocol producing random outcomes using imperfect sources of randomness — albeit using a different definition than considered here. The initial idea was proposed by Colbeck and Renner [20] with a construction involving chained Bell inequalities[35]
for 2 parties, albeit with a minimal threshold of independence. Exact thresholds were computed by Grudka et al. [36], while subsequent works [37,38,39,40,41,42]
proposed protocols which are able to amplify arbitrarily weak initial randomness.
These constructions are all based on common Bell inequalities.
In 2014, a breakthrough was realized by our colleague Gilles Pütz [7], and we reproduce below, with his permission, the result of our subsequent collaboration.
Scenario for measurement-dependent locality. — We consider a scenario where two parties, Alice and Bob, have each access to a box. They provide mea- surement settings as an input to their box, denoted by the random variables X (respectively Y for Bob), and interpret the output of the box as their measurement outcome, with associated random variable A (respectively B).
Alice and Bob perform the experiment many times, and register each time the measurement settings (x, y) selected at random and the measurement outcomes (a, b) received (in this Chapter we use capital letters for random variables, and lower case letters for the values these random variables can take). From the number of events corresponding to (a, b, x, y), the parties can estimate the probability
2.2 Weakened measurement independence 25
PA B XY(abx y) of drawing measurement settings (x, y) and receiving measurement outcomes (a, b).
Note:in the literature assuming perfect measurement independence, it is customary to use the conditional probability distribution PA B|X Y(ab|xy)instead of PAB X Y. Indeed, whenX andYare random variables totally independent from the rest of the system, no information can be gained from the distribution PX Y(x y). On the contrary, observing that X and Y are not independent
— PX Y(xy) =/ PX(x)PY(y)— would draw suspicions in an experiment. It is thus worthwhile to preserve information about PX Y.
When the scenario uses quantum resources, each box receives part of a quantum system, and the inputsX andY select the quantum measurement performed on the state by the boxes.
In a local scenario, the quantum system is replaced by a random variableΛ(with valueλ), and the measurement resultAdepends onXandΛ, while the measurement result B depends onY and Λ. The inputs X and Y themselves are chosen freely:
PXY|Λ(xy|λ) =PXY(xy). (2.1) Following [29, 31], we call this assumption measurement independence. With this assumption, correlations PAB XY are local if there exist distributions ρΛ, PA|XΛ, PB|YΛsuch that:
PA B XYlocal (abxy) =PXY(xy) Z
dλ ρΛ(λ)PA|XΛ(a|xλ)PB|YΛ(b|yλ). (2.2) On the other hand, when measurement independence is relaxed, the distribution of the inputs X,Y depends on Λ. We define another model, such that correlations PA B XY are measurement dependent local (MDL) if:
PA B XYMDL (abxy) = Z
dλ ρΛ(λ)PX Y|Λ(xy|λ)PA|XΛ(a|xλ)PB|YΛ(b|yλ). (2.3)
a b
X Y
x y
a b
x y
a) Bell locality b) measurement dependent locality
Figure 2.3. Bell local and measurement dependent local scenarios.
If no further assumptions are made, we already remarked that nonlocality cannot be demonstrated in deterministic scenarios. We thus restrict measurement depen- dence using lower and upper bounds ℓ, h:
ℓ6PXY|Λ(xy|λ)6h. (2.4)
Correlations are then measurement dependent nonlocal (MD nonlocal) for given (ℓ, h) if they cannot be expressed as (2.3) while respecting (2.4). Scenarios cor- responding to Bell locality and measurement dependent locality are depicted in Figure 2.3.
26 Introduction to independence condtions
We can now state the main result of [7]: there exist quantum correlations that are measurement dependent nonlocal for ℓ >0 and any h. This observation is based on a new inequality satisfied by all MDL correlations. We then show the existence of quantum correlations violating the inequality for ℓ >0.
Inequality for MDL correlations. — In a scenario with binary inputs and outputs (a, b, x, y= 0,1), the following inequality is verified by all MDL correlations with MDL bounds (ℓ, h):
I=ℓP(0000)−h[P(0101) +P(1010) +P(0011)]MDL≤ 0. (2.5) Quantum violation. — This inequality can be violated for any ℓ >0, for example using the state:
|Aui= 1
√3 √5
−1
2 |00i+√5 + 1 2 |11i
!
, (2.6)
and measurements:
|Axi=cosθx|0i+sinθx|1i, |Byi=cosθy|0i −sinθy|1i, (2.7) with θ0=cos−1 12+ 1
√5
q ,θ1=θ0−π4.
Evaluating the left-hand side of inequality (2.5) for those quantum resources, we find I=ℓ121 PXY(00). When ℓ >0, the probability PXY(00) of choosing the inputs (x, y) = (0,0) is nonzero, and thus the inequality is violated whenever ℓ >0. This means that the local hidden variableΛcan influence the selection of the inputs(x, y) as long as all pairs have an ℓ >0probability of being chosen for any λ.
A geometrical picture of the result is presented in Figure 2.4.
Nonsignaling distributions Quantum correlations CHSH inequality CHSH shifted for Our MDL inequality for
Quantum violation for Behavior when
a b
x y
Figure 2.4. Adapted from [7], slice of the correlation space. The quantum correlations (dotted) are a subset of the nonsignaling correlations (in white). The quantum point corresponding to the state and measurements in (2.6) is represented by a white dot. This figure presents a comparison of different inequalities: the CHSH inequality without shift is invalid, as the measurement independence condition is not respected. When shifting the CHSH inequality to take into account a measurement dependence characterized by ℓ= 2−√2
/4, the inequality no longer exhibits quantum violations. However, our MDL inequality is tailored for the situation and shows a violation for anyℓ >0, rotating around the point in black as needed. Arrows indicate the movement of the shifted CHSH inequality and our MDL inequality for increasing ℓ.
2.2 Weakened measurement independence 27
Other results. — Other results were presented in [7], for correlations in a slice obeying two observations. First, all inputs are observed to appear with the same probability (PXY(xy) =14 is constant). Second, the conditional distributionPA B|XY
is nonsignaling. The fulfilment of those conditions is sought in experiments: nobody would use a visibly biased random number generator, and special relativity along with space-like separation of the devices guarantees nonsignaling. Having PXY(xy) constant simplifies greatly the handling of Bell-like inequalities: a constant factor relates the distribution PA BXY of the MDL models and the conditional distribution PA B|XY used in the literature.
Having related the MDL correlations with conditional distributions used in Bell tests, the paper also presents MDL violations of the CHSH inequality, and showed that for ℓ 6 2− 2
√
4 or 1 − 3h 6 2− 2
√
4 MDL correlations can violate the CHSH inequality, in contrast with the inequality (2.5). This is not surprising, as the coef- ficients of CHSH are derived with the assumption of measurement independence, while the MDL framework describes precisely the effects of bounded measurement dependence.
Measurement dependence can also be modeled by min-entropy sources, com- monly studied in the literature [32]. These sources obey the constraint (2.4) with ℓ = 0 and h > 0. The polytope of MDL correlations obeying these additional constraints forℓ= 0and h∈1
4,13
has its boundary described by7 families of Bell- like inequalities (the limiting cases h = 14, 13 fall back on local and nonsignaling theories respectively).
Note:a small assumption was made about the polytope keeping its algebraic structure invariant under continuous deformation. Let the algebraic structure of the polytope be described by its inci- dence matrix. The following is left as an open question: let {~vk(t)}be vertices of a polytope linear in the parametert. Lett0andt1be two values oft such that the polytope has the same incidence matrix (describing its structure) fort0andt1. Is the incidence matrix of the polytope then constant in the interval [t0, t1]?
2.3 Conclusion
The concept of locality developed by Bell requires an unfalsifiable assumption, mea- surement independence. This assumption can be replaced by another assumption directly derived from the locality principle, namely n-locality: independent phys- ical systems are described by uncorrelated local hidden variables. In the process, we described Bell tests without inputs, where measurement settings are chosen by independent random number generators.
In collaboration with Gilles Pütz and other colleagues, we studied the case where this assumption is relaxed, and showed that quantum nonlocality can still be demon- strated even in presence of arbitrarily small measurement independence. The demon- stration was made possible by a direct characterization of the set of measurement dependent correlations, in contrast with previous works that instead corrected the bounds of existing Bell inequalities.
28 Introduction to independence condtions
Chapter 3
Bilocality and entanglement swapping
The n-locality assumption is particularly adequate to describe correlations in sce- narios containing multiple independent sources. Consider, for example, the scenario of entanglement swapping, where a quantum state ρAB is distributed to the devices A and B, and another state ρBC to the devices B and C, as drawn in Figure 3.1.
Interpreting now local hidden variables as attached to the states under measurement, we letΛABandΛBCdescribe the statesρABandρBCin the corresponding local model.
In the event that ρABand ρBCare produced using physical devices that have nothing in common, it is natural to assume that ΛAB and ΛBC are independent random variables — in any case, it follows the spirit of the measurement independence condition present in Bell’s theorem. We name this new assumption the bilocality, or 2-locality assumption.
What are the consequences of assuming the independence of local hidden vari- ables distributed in a correlation scenario? As the model has additional constraints compared to Bell locality, its correlation set should be smaller. For a given quantum point, its distance to the 2-local set is equal or greater than its distance to the Bell-local set of correlations. It is thus expected, at least for part of the quantum correlation set, that the new models would lead to stronger violations of locality.
However, how should the strength of locality violations be measured? The first road is to consider experimental imperfections, such as noise in the states, or inef- ficient detectors, and see how much we can deviate from an ideal scenario before the demonstration of nonlocality breaks down. The second road is to compute the minimal amount of additional resources needed, on top of the local hidden vari- ables, to simulate the nonlocal correlations. In this Chapter, we demonstrate the robustness of nonlocality in the presence of imperfections. We review in Chapter 4 several models simulating the correlations of entanglement swapping scenarios using
BS
PPLN waveguide
Source 1 Source 2
Bragg filter
Continuous-wave laser
APD APD
SSPD
Si filter
BSM APD
+ –
+ –
A
x
a
B
b
C
z
c
x z
Source 1 Source 2
(1) (2) (3) (4) (1) (2) (3) (4)
Figure 3.1. Entanglement swapping experiment by Halder et al.[43], along with its causal structure. Figure adapted by permission from Macmillan Publishers Ltd: Nature Physics [43], (c) 2007.
29
bounded classical communication in addition to local hidden variables.
Our central question is thus: can we demonstrate stronger quantum violations of locality using a bilocal interpretation of the locality principle, compared to Bell locality? More generally, can we characterize, at least partially, the set of bilocal cor- relations, and find generalizations of Bell inequalities to parameterize the boundary of the set? We give affirmative answers to both questions in this Chapter.
The scenario considered is entanglement swapping [44], where two entangled states, created independently in sites 1-2 and 3-4, can manifest entanglement in 1-4 after a joint measurement in 2-3. Then, the particles in 1 and 4 will exhibit correlations, even if those particles have never interacted, nor have anything in common in their past (see Figure 3.1).
Previous results. — It was observed by Żukowski et al. that conditioned on the result of a joint measurement, entanglement can be swapped and tested in remote particles [44]. The first local hidden variable models adapted for entangle- ment swapping were proposed by Gisin and Gisin in [45], Greenberger et al. in [46, 47], albeit with additional restrictions on top of bilocality. A breakthrough was made by Branciard et al. in [22], constructing the first nonlinear inequality for a bilocal scenario, and demonstrating stronger quantum violations after addition of the bilocality constraint.
Our contributions. — The previously known inequality required a complete Bell measurement, as well as perfectly efficient detectors. In our work, we demonstrate that stronger violations of quantum nonlocality still exist when several variants of partial Bell measurements are considered. We extended the previous result in the case the two extremal parties have inefficient detectors, and shows that bilocality still offers an advantage over locality.
On a fundamental level, we gave a complete mathematical description of the set of bilocal correlations, while providing examples of nonlinear Bell-like inequalities that are tight in specific slices of the correlation space, offering necessary but not sufficient conditions everywhere else.
Most of the work presented was published in [1]; in addition, we provide in Appendix A a previously unpublished analytical proof of the result presented in [1]
concerning the case of inefficient detectors.
3.1 A bilocal scenario for entanglement swapping
We consider three parties, Alice, Bob and Charlie. Alice and Charlie have binary inputs x, z= 0,1 and ±1-valued binary outputs A, C =±1, whereas Bob input is written y, and its output is writtenB= (B1...BL)∈ {±1}L— to allow him to output L bits. A first sourceSABis connected to Alice and Bob, and a second sourceSBCis connected to Bob and Charlie. We describe the correlations in the scenario by the probability distribution P(ABC|xyz).
Quantum correlations. — To move towards an implementation of our bilocal tests using linear optics, we consider variants of Bell measurements. Indeed, com- plete Bell measurements on two photons (able to project on the four Bell states) are not possible using linear elements [48, 49].
30 Bilocality and entanglement swapping
The first variant is a partial Bell measurement where Bob choose whether he wants to know if the projected state is: |Φ±i vs. |Ψ±i, or |Φ+/ Ψ+i vs. |Φ−/ Ψ−i, corresponding to an input y= 0,1and a bit of outputB=±1. Surprisingly, the first question (Φvs.Ψ) can be answered by the measurementB0=σz⊗σz, and the second question (+vs. −) by the measurementB1=σx⊗σx, both separable measurements that can be realized using linear optics. Let us observe that these measurements commute [B0, B1] = 0, and when measured together implement a complete Bell measurement.
The second variant is a partial Bell measurement where Bob can discriminate between |Φ+i, |Φ−i and |Ψ+/−i, a measurement that again can be implemented using linear optics. In this variant, Bob does not have to choose an input, and, for simplicity, his measurement outcome is encoded in two bitsB0B1=00,01,{10 or 11}. Alice and Charlie perform projective measurements on the qubits they share with Bob, and the states ρAB,ρBCdistributed between the parties are singlets mixed with white noise (also known as Werner states [50]):
ρAB=vAB|Ψ−ihΨ−|+ (1−vAB)|Ψ−ihΨ−|, (3.1) ρBC=vBC|Ψ−ihΨ−|+ (1−vBC)|Ψ−ihΨ−|. (3.2) Local and bilocal models. — When the correlationsP(ABC|xyz) are bilocal, there exist a bilocal model with random variables λAB ∈ ΛAB, λBC ∈ ΛBC with probability densities ρAB(λAB), ρBC(λ) such that:
P2-LOC(ABC|xyz) = Z
λAB∈ΛAB
dλAB
Z
λBC∈ΛBC
dλBCρAB(λAB)ρBC(λBC)·
·PA(A|xλAB)PB(B|yλABλBC)PC(C|zλBC).
(3.3)
In contrast, a model respecting Bell locality is given by:
PLOC(ABC|xyz) = Z
λ∈Λ
dλ ρ(λ)PA(A|xλ)PB(B|yλ)PC(C|zλ), (3.4) a difference illustrated in Figure 3.2.
A
x
A
B
B
C
z
C y
Bilocal model Local model
A
x
A
B
B
C
z
C y
Figure 3.2. Comparison of the distribution of variables for bilocal and local models of entanglement swapping experiments.
3.1 A bilocal scenario for entanglement swapping 31
