Quantum entanglement and measurement incompatibility as resources for nonlocality

Thesis

Reference

Quantum entanglement and measurement incompatibility as resources for nonlocality

QUINTINO, Marco Tulio

Abstract

Cette thèse étudie la connexion entre lintrication quantique, lincompatibilité des mesures quantiques, le steering d'Einstein-Podolsky-Rosen et la nonlocalité de Bell. Nous développons des méthodes générales pour travailler avec les mesures quantiques généraux (Positive Operator Valued Measurement) et répondons à des questions fondamentales sur la mécanique quantique.

QUINTINO, Marco Tulio. Quantum entanglement and measurement incompatibility as resources for nonlocality . Thèse de doctorat : Univ. Genève, 2015, no. Sc. 4979

URN : urn:nbn:ch:unige-880930

DOI : 10.13097/archive-ouverte/unige:88093

Available at:

http://archive-ouverte.unige.ch/unige:88093

Disclaimer: layout of this document may differ from the published version.

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Université deGenève Faculté desSciences Département de Physique Théorique Professeur N. Brunner

Quantum Entanglement and Measurement Incompatibility as

Resources for Nonlocality

THÈSE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès

sciences, mention physique par

Marco Túlio Quintino

de

Divinópolis (Brésil) Thèse No4979

Genève

Atelier Repromail, Université de Genève 2016

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Czas – mój najwi˛ekszy wróg, mój na- jlepszy przyjaciel

Czas – nie u˙zywa słów, ale zawsze odnajdzie

Closterkeller - Na Kraw˛edzi

Abstract

The subject of this thesis is understanding the connection between quantum en- tanglement, quantum incompatible measurements, Einstein-Podolsky-Rosen- steering and Bell nonlocality. The concept of nonlocality allows one to discuss some distinctive features of quantum theory from an external and general perspective. With this approach one can contrast of quantum mechanics with classical physics and even more general ones.

In the EPR/Bell nonlocality framework, we recognise quantum entangle- ment and incompatible quantum measurements not only as two concepts lying at the core of quantum theory, but also as two central ingredients to show that quantum theory is not a Bell local causal one. While compatible measurements and separable states can never lead to nonlocal quantum correl- ations, the converse question is more subtle. In this thesis we make progress in understanding which entangled states and which sets of incompatible measurements can (and cannot) lead to EPR/Bell nonlocality in a bipartite scenario. We consider general measurements,i.e., Positive Operator Valued Measures, and general mixed entangled states. We show that although the assumption of projective measurements can greatly simplify some problems, it is possible to develop some methods to tackle general POVMs and learn some fundamental aspects of quantum theory.

The first part of the thesis focuses on quantum states. We show that, even when the parties have access to general POVMs, there exist entangled quantum states that cannot lead to EPR-steering, and moreover there exist EPR-steerable states that cannot lead to Bell nonlocality. Rigorously, this proves the inequivalence of entanglement, EPR steering, and Bell nonlocality.

We also prove the existence of a fundamental asymmetry of EPR-steering in quantum mechanics, the existence of states for which Alice can steer Bob, but Bob can not steer Alice. We obtain our results by developing general methods that can be used to construct states with the desired property.

The second part of this thesis focuses on quantum measurements. We start by showing that every set of incompatible quantum measurements can lead to EPR-steering. That is, for any set of incompatible measurements there exists an entangled quantum state such that the assemblage held by the other part is steerable. This direct connection between measurement incompatibility and EPR-steering is made in a constructive way and we discuss direct applications in both fields. We then show that this strong connection between measurement compatibility and EPR-steering cannot be extended to Bell nonlocality. This is demonstrated by presenting a set of incompatible measurements that when

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performed by Alice, it can only lead to Bell local statistics, regardless the quantum state shared by the parties and general dichotomic measurements that can be performed by Bob.

Finally, in the third part of the thesis we investigate an EPR-steering scenario where the parties share multiple copies of a bipartite quantum states.

We present a general criterion to detect entangled states that lead to EPR- steerable statistics when the parties can perform joint measurements on k copies of it. Our criterion is then exploited to prove the existence of super- activation of EPR-steering. That is, we show that some entangled states which admit a Local Hidden State model, hence useless for EPR-steering in the standard single-copy regime, can become steerable when more copies are considered. Moreover, we present a general method to tackle the two- copy case, that is, constructing statesρ_ABwhich are EPR-unsteerable (even with general POVMs) but ρ_AB⊗ρ_AB is steerable. We also contrast super- activation of EPR-steering with the super-activation of Bell nonlocality. Some particularities of EPR-steering allow us to greatly enlarge the previous set of states that can have their Bell nonlocality super-activated. In particular, we show that all entangled two-qubit and qubit-qutrit states are EPR-steerable when sufficiently many copies are considered.

Résumé

Cette thèse étudie la connexion entre l’intrication quantique, l’incompatibilité des mesures quantiques, le steering d’Einstein-Podolsky-Rosen et la non- localité de Bell. Le concept de nonlocalité nous permet de discuter des particularités de la mécanique quantique avec une perspective externe et plus générale, ce qui nous amène à contraster la mécanique quantique avec la physique classique et également avec des théories plus générales et abstraites.

Dans le cadre de la nonlocalité EPR/Bell, l’intrication quantique et l’incompa- tibilité des mesures, deux conceptscentraux de la théorie quantique, sont aussi deux ingrédients fondamentaux pour montrer que la théorie quantique est nonlocale au sens de Bell. Tandis que les mesures compatibles et les états séparables ne peuvent jamais produire de statistique nonlocale, la question inverse est plus subtile.

Cette thèse reporte des progrès dans la compréhension de quels états intriqués et quels ensembles de mesures incompatibles peuvent (et ne peuvent pas) produire de nonlocalité EPR/Bell dans un scenario à deux parties. Un effort particulier est devué à l’examen des mesures générales, c’est-à-dire„

Positive Operator Valued Measure, et des états mixtes généraux.

Nous montrons que, même si l’hypothèse de mesures projectives peut grandement simplifier les problèmes reliés aux mesures quantiques, il est quand même possible de développer des méthodes générales pour travailler avec les POVMs et répondre à des questions fon- damentales sur la mécanique quantique.

La première partie de cette thèse est dédiée aux états quantiques. Nous prouvons que, même si les POVMs généraux sont considérés, il existe des états intriqués qui ne sont pas EPR-steerables, des états qui sont EPR-steerables mais pas Bell nonlocaux. Nous montrons aussi l’existence d’une asymétrie fondamentale du EPR-steering au niveau des états quantiques en présentant des états pour lesquels Alice peut steerer Bob mais Bob ne peut pas steerer Alice. Ceci démontre formellement équivalence entre intrication, EPR-steering, et la nonlocalité de Bell. Les démonstrations présentés sont basées sur des méthodes générales qui peuvent être utilisées pour construire un état avec la propriété désirée.

La deuxième partie de la thèse se concentre sur les mesures quantiques.

Nous prouvons que tous les ensembles de mesures incompatibles peuvent produire des corrélation de type EPR-steering. Formellement, nous montrons que pour tout ensemble de mesures incompatibles il existe un état intriqué tel que si Alice effectue ces mesures sur cet état, l’assemblage de Bob sera

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steerable. Cette connexion est établie de façon constructive et nous permet de établir un lien entre ces deux domaines. Ensuite, nous montrons que ce lien étroit entre l’incompatibilité des mesures et le EPR-steering ne se tien pas dans le cadre de la nonlocalité de Bell. Nous construisons un ensemble de mesures incompatibles qui, si effectué par Alice, ne peut jamais générer de statistiques nonlocales, indépendamment de l’état partagé par les parties et pour toutes les mesures dichotomiques que Bob peut effectuer.

Finalement, dans la la dernière partie de la thèse, nous étudions l’EPR- steering dans un scenario où les parties ont accès à plusieurs copies de leur état partagé. On présente un critère général pour détecter les états intriqués qui sont steerables quand les parties peuvent effectuer des mesures jointes sur k copies d’un état. Ensuite, on utilise ce critère pour prouver l’existence d’une super-activation du EPR-steering. Nous montrons qu’il existe des états intriqués qui admettent un modèle local à état caché (Local Hidden State model), donc inutiles pour l’EPR-steering dans un régime de copie unique, mais qui deviennent steerable dans le régime de plusieurs copies.

En outre, nous construisons des étatsρ_ABqui sont EPR-unsteerable (même avec des POVMs généraux), tel que l’étatρ_AB⊗ρ_AB sont steerables. Nous comparons aussi la super-activation du EPR-steering avec la super-activation de la nonlocalité de Bell. Ceci met en évidence les particularités du EPR- steering qui nous ont permis d’augmenter considérablement l’ensemble des états qui peuvent avoir leur nonlocalité super-activée. En particulier, nous prouvons que tous les états intriqués de deux qubits ou qubit-qutrit sont EPR-steerables quand un nombre suffisamment grand de copies est considéré.

Remerciements

The results presented in this thesis would never have been obtained without all the help and support of many friends and collaborators. Innumerous people have contributed directly and indirectly for this PhD. The number of contributions and important discussions is so big that I do not believe I can list all names in just a couple of pages. Aware that I should restrict myself just to a few names, I will now enjoy the pleasure of acknowledging some friends and collaborators being sure that some very important names will be forgotten.

Je voudrais tout d’abord remercier Nicolas Brunner. Merci pour avoir partagé toute ta créativité et tes nombreuses idées. Merci pour m’avoir aidé à organiser mes présentations (orales et écrites). Merci pour m’avoir appris comment bien écrire un article et un grant application. Merci pour fournir un environnement de travail super agréable. Merci aussi pour toutes les conversations dehors de la physique. Tu es beaucoup plus qu’un directeur de thèse pour moi. Merci pour tout!

Un immense merci pour mes deux grands amis et collaborateurs Joe et Flavien! Une grande partie des résultats de mon doctorat est une conséquence directe de nos discussions sur la physique et de nos discussions éternelles sur

“la définition de la définition’.

Não poderia me esquecer do grande Marcelo Terra. Obrigado por ter me acolhido ainda nos tempos de Cálculo1. Obrigado por todo conhecimento, toda base, e todas as oportunidades que você me proporcionou e ainda proporciona! E claro, ao me lembrar dos tempos de mestrado, é impossível não pensar no meu grande amigo e colaborador Mateus. Valeu Mateus, ainda temos muitos trabalhos juntos a fazer!

Um enorme obrigado ao Daniel. Além de sua grande importância para minha formação em não-localidade e nas primeiras publicações em física, foi quem me sugeriu trabalhar com o Nicolas e possibilitou o começo deste doutorado. Valeu daniel!

Um grande obrigado também a todos terráquios que tiveram e ainda têm um impacto forte que não se restringe aos trabalhos em física, Gláucia, Rafael, Leo, Jessica, Dutty e toda turma! Obrigado também ao "pessoal da UFMG":

Tche, Planeta, Fernando, Marcelo (França).

Thank you Tamás for countless physics discussions and great times. It is always a big pleasure to work and talk to you. Köszönöm szépen!

I would also like to thank Nicolas Gisin, Jonatan Brask, Tomer Barnea, Denis Rosset, Toni Acín, Paul Skrzypczyk, Matthew Pusey.

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Contents

Introduction 13

1 Quantum Mechanics and Nonlocality 16

1.1 The CHSH game . . . 16

1.2 A Framework for Bell Nonlocality . . . 18

1.2.1 Bell inequalities . . . 20

1.2.2 Fine’s Theorem: Another Way to Understand Bell Non- locality . . . 21

1.2.3 The Non-Signalling Paradigm . . . 22

1.3 Quantum Nonlocality . . . 23

1.3.1 Quantum States. . . 23

1.3.2 Quantum Measurements . . . 24

1.3.3 Quantum Entanglement . . . 24

1.3.4 Bell’s Theorem . . . 25

1.4 EPR-steering . . . 26

1.5 Entangled Local States . . . 29

1.5.1 Werner States . . . 31

1.5.2 Isotropic States . . . 32

1.5.3 More general States . . . 33

1.5.4 A General Algorithm To Construct LHV/LHS Models . 34 2 Inequivalence of Bell Nonlocality, Quantum Entanglement, and EPR-Steering 35 2.1 Quantum EntanglementvsEPR-steering . . . 36

2.1.1 Constructing POVM LHV/LHS models from projective ones . . . 37

2.1.2 LHS Models Based On Local Filtering Operations . . . . 39

2.2 EPR-steeringvsBell Nonlocality . . . 41

2.3 One-WayvsTwo-way EPR-Sterring. . . 42

2.4 Hidden Steering. . . 44

2.5 Final Remarks . . . 45

3 Measurement IncompatibilityvsEPR-steering 46 3.1 Measurement Incompatibility in Quantum Mechanics. . . 47

3.2 Equivalence Between Measurement Compatibility and EPR- Steering. . . 48

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CONTENTS 10

3.3 Applications Of The Correspondence Between Joint Measurab-

ility And EPR-Steering . . . 49

3.3.1 Applying Results From EPR-steering to Measurement Incompatibility . . . 49

3.3.2 Applying Results From Measurement Incompatibility To EPR-steering . . . 50

3.3.3 A direct correspondence Between Joint Measurability and EPR-Steering . . . 53

3.4 Final Remarks . . . 54

4 Measurement Incompatibilityvs Bell Nonlocality 55 4.1 Inequivalence Between Measurement Compatibility and Bell Nonlocality . . . 56

4.2 Hollow Triangle Incompatibility And Bell Nonlocality . . . 59

4.2.1 Technical Methods . . . 62

4.3 Final Remarks and Future Directions . . . 62

5 Super-Activation of Quantum Steering 64 5.1 The Many Copies Paradigm On Bell Nonlocality. . . 64

5.2 Super Activation of Bell Nonlocality . . . 65

5.3 Super Activation of EPR-Steering . . . 67

5.3.1 All Entangled Two-Qubit States arekCopy Steerable . . 69

5.3.2 The Case of Asymmetric Local Dimension . . . 70

5.3.3 An Alternative Proof For Super-Activation of Quantum Steering . . . 70

5.4 Requirements on Dimension and Number of Copies For Super Activation . . . 72

5.5 Super Activation of Nonlocality And Entanglement Distilation 73 5.5.1 Super Activation of EPR-steering and One-Way Distillation 73 5.6 Final Remarks and Future Directions . . . 74

Conclusions and perspectives 75

List of Publications During The PhD 78

Bibliography 80

A Papers published in the course of this thesis 89

List of publications

Chronological list of papers published during the PhD.

• [MTQ1]Genuine Hidden Quantum Nonlocality F. Hirsch, M. T. Quintino, J. Bowles, and N. Brunner Physical Review Letters111,16(2013)160402

Mentioned in chapter2and section1.5.3.

• [MTQ2]Certifying the Dimension of Classical and Quantum Systems in a Prepare-and-Measure Scenario with Independent Devices J. Bowles, M. T. Quintino, and N. Brunner

Physical Review Letters112,14(2014)140407 Paper not discussed in this thesis.

• [MTQ3]One-way Einstein-Podolsky-Rosen Steering J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner Physical Review Letters112,20(2014)200402

Mentioned in chapter2and section1.5.3.

• [MTQ4] Joint Measurability, Einstein-Podolsky-Rosen Steering, and Bell Nonlocality

M. T. Quintino, T. Vértesi, and N. Brunner Phys. Rev. Lett.113,16(2014)160402, Paper covered in section3.

• [MTQ5]Local Hidden Variable Models for Entangled Quantum States Using Finite Shared Randomness

J. Bowles, F. Hirsch, M. T. Quintino, and N. Brunner Physical Review Letters114,12(2015)120401,

Partially discussed in section1.5.4

• [MTQ6] Inequivalence of entanglement, steering, and Bell nonlocal- ity for general measurements

M. T.. Quintino, T. Vértesi, D. Cavalcanti, R. Augusiak, M. Demianowicz, A. Acín, and N. Brunner

Phys. Rev. A92, (2015)032107 Paper covered in section2.

• [MTQ7]Sufficient criterion for guaranteeing that a two-qubit state is unsteerable

J. Bowles, F. Hirsch, M. T. Quintino, and N. Brunner 11

CONTENTS 12 Phys. Rev. A93,2(2016)052115,

Paper mentioned in chapter4and section1.5.3.

• [MTQ8]Incompatible quantum measurements admitting a local hid- den variable model

M. Túlio Quintino, J. Bowles, F. Hirsch, and N. Brunner Phys. Rev. A93, (2016)032107

Paper covered in section4.

• [MTQ9]Algorithmic construction of local hidden variable models for entangled quantum states

M. T.. Quintino, T. Vértesi, D. Cavalcanti, R. Augusiak, M. Demianowicz, A. Acín, and N. Brunner

Phys. Rev. A92, (2015)032107 Paper mentioned in section1.5.4.

• [MTQ10]Entanglement without hidden nonlocality

F. Hirsch, M. Túlio Quintino, T. Vértesi, M. F. Pusey, and N. Brunner arXiv:1606.02215 [quant-ph]₍²⁰⁰⁶₎

Paper mentioned in section1.5.4.

• [MTQ11]On the tightness of correlation inequalities with no quantum violation

R. Ramanathan, M. Túlio Quintino, A. Belén Sainz, G. Murta, and R. Au- gusiak

arXiv:1607.05714 [quant-ph]₍²⁰¹⁶₎ Paper not discussed in this thesis.

Introduction

Philosophy and physics are two branches of research that always walk to- gether. Philosophy provides arguments, reasoning and motivations for physics and physics complements this relation by providing solid mathematics and rigorous scientific methods to tackle deep philosophical questions. Boosted by the emergence of quantum mechanics, many physicists of the beginning of the20th century asked themselves questions like, “What is reality?”, “What is randomness?”, “Do we really need probabilities to describe nature?”

Famous for being counter-intuitive, quantum mechanics provides a very accurate description of nature. Its predictions are confirmed experimentally up to great precision and its impact on the technology is undeniable. Differently from Newtonian physics and classical statistical mechanics, quantum theory features intrinsically random predictions at the most fundamental level. This intrinsic randomness bothered many physicists since quantum mechanics was mathematically formalised and was one of the main motivations for the questions mentioned above.

A first important step on understanding this probabilistic nature of quantum theory was made by Einstein, Podolsky, and Rosen when they addressed the question “Can Quantum-Mechanical Description of Physical Reality be Con- sidered Complete?”. In reference [1], they exploited quantum systems that were spatially separated to propose a possible paradox between quantum mechanics and the assumption that an isolated party cannot have an instantan- eous causal influence on the other. The result presented by Einstein, Podolsky, and Rosen became known under the name ofEPR paradox, and just after it was published, Schrödinger pointed out the existence ofquantum entanglement as one main ingredient of it [2].

A breakthrough in understanding the EPR paradox was made by Bell in 1964[3]. Bell tackled the problem first by formalising alocal causal theoryin two main assumptions:1– There exists a variable, which we may not have access to, that can be used to predict the outcome of measurements with probability one. 2– Operations on different isolated parts cannot instantaneously have any causal influence on the other. Then, Bell showed that the predictions made by quantum mechanics for some entangled states are not compatible with any local causal theory and this phenomenon is known under the name ofquantum nonlocality. Since Bell, the connection between entanglement and nonlocality has been intensively studied. The definition of entanglement may suggest that all entangled states are nonlocal, but in1989, Werner showed that certain entangled states cannot lead to nonlocal correlations [4], showing that

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CONTENTS 14 the connection between entanglement and nonlocality may be very subtle.

In2007Wisemanet alrevisited the EPR paradox through a more modern view of quantum mechanics. In reference [5] they formalised the gedanken- experiment proposed by Einstein, Podolsky, and Rosen to obtain the notion of EPR-steering. EPR-steering can be understood as a nonlocal phenomenon lying in between Bell’s nonlocality and quantum entanglement. Entanglement is a concept that addresses directly quantum states. Bell nonlocality can be applied for any general physical theory. EPR-steering has a intermediate balance between assuming quantum mechanics and general physical theories.

Another fundamental and necessary ingredient for nonlocality is the ex- istence ofincompatible measurements, another concept that lies at the core of quantum mechanics. The fact that measuring the momentum of a particle necessarily disturbs its position was one of the pillars for the formalisation of quantum theory [6]. Historically, the connection of measurement incompatib- ility and nonlocality was less explored than its connection with entanglement.

We mention here two important results: in1986Khalfin and Tsirelson noticed that all projective incompatible measurements can lead to quantum nonloc- ality [7,8] and2009 by Wolf and collaborators showed that every pair of dichotomic incompatible measurements can lead to Bell nonlocality [9]. This gave evidence of a possible strong connection between Bell nonlocality and measurement incompatibility.

The main goal of this thesis is to understand how quantum entanglement and quantum measurement incompatibility relate to nonlocality. The novel results we present here were mainly motivated by four fundamental questions:

Chapter2: Which entangled quantum states can lead to Bell nonlocality and EPR-steering?

Chapter 3: Which sets of quantum measurements can lead to EPR- steering nonlocality?

Chapter 4: Which sets of quantum measurements can lead to Bell nonlocality?

Chapter5: Canρ⊗ρbe Bell/EPR nonlocal ifρis Bell/EPR local? i.e., can nonlocality besuper activated?

We start the thesis by introducing some basic concepts on nonlocality and quantum mechanics in chapter1and then move to our contributions.

In chapter2we analyse quantum entangled states that when submitted to general quantum measurements,i.e., Positive Operator Valued Measure, cannot lead to EPR-steering and Bell Nonlocality. We show that even when general quantum measurements are considered, entanglement, EPR-steering, and Bell nonlocality are three different concepts in quantum mechanics.

In chapter3we show that every set of quantum incompatible measure- ments can lead to EPR-steering correlations in a bipartite scenario where the parties share a pure entangled state. This establishes a strong correspondence between EPR-steering and joint measurability which leads to several results in each field.

CONTENTS 15 In chapter 4 we show that there exists a set of incompatible Khalfin measurements that can only lead to Bell local correlations. More precisely, if one party performs this set of incompatible measurements, for any possible shared entangled state, and any possible dichotomic measurements performed by the second part, the resulting statistics are Bell local.

In chapter5we analyse an EPR-steering scenario where the parties share many copies of an specific state. We provide a simple general method for Alice to steer Bob when they sharekcopies of some entangled stateρ. Our method can be applied for a large class of quantum entangled states, including all entangled two-qubit states and all qubit-quitrit ones.

Chapter 1

Quantum Mechanics and Nonlocality

In this chapter we introduce some basic concepts of quantum mechanics and nonlocality that will be explored in the rest of the thesis. Although the main definitions will be covered, the introduction presented here is brief.

Readers that do not feel comfortable with quantum mechanics and quantum information are invited to read the book of Nielsen and Chuang [10] and the lecture notes of John Watrous [11]. For a recent review on nonlocality we recommend reference [12]. We also mention the Horodecki family review on quantum entanglement [13], Gühne and Tóth review on entanglement detection [14], and Augusiaket al.review on quantum states admitting a local hidden variable model [15].

1.1 The CHSH game

The art of doing mathematics is finding that special case that contains all the germs of generality.

David Hilbert

Alice and Bob, two good old friends, are invited by a referee to play the CHSH game[16]. The referee proposes10pieces of gold for each one in case they manage to satisfy somewinning conditions. Alice and Bob liked the idea and start asking a few questions.

Alice: So, how does this game work? Do we both win together?

Referee: Yes! In case thewinning conditionis satisfied, both win. If not, you do not receive anything.

Bob: Hum... but what are thesewinning condition? How does the game work?

Referee: Ok, let me explain. I will send each one an “input bit”, that is a simple technical term for a small card with the number 0 or the number 1 written on it. Also, both inputs are going to be chosen in an independent and uniform way.

Alice: I see...

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CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 17 Referee: For convenience, Alice’s card labelled asx, and in case the card has the number 0 written on it, we say thatx=0, similarly, if the card has the number 1, we sayx=1.

Bob: And for me?

Referee: Bob’s card is labelled byy. Similarly to Alice, we say thaty=0, when the card reads 0 andy=1 if the card reads 1.

Alice: Ok! And what should we do with these cards?

Referee: After receiving your inputs, each one should return thoseoutput bitsto me. That is simply a card with zero or one written on it. Alice’s output is labelled asa, and Bob’s labelled asb.

Bob: Humm. . . So the only thing we should do is to wait for an input, choose an output and send it to you?

Referee: Yes!

Alice: And what are thesewinning conditions?

Referee: After receiving your outputs, I will analyse the relation of inputs and outputs. Whenever you receive the combination (x=0,y=0), (x=0,y=1),(x=1,y=0), you win when the outputs are the same, that when a = b. The case(_x = _1,y = ₁) is special! If both of you receive input one, the outputs should be the opposite, that is,a6=b.

Bob: Ok, sounds simple.

Referee: There is also one more rule. Just before receiving the inputs each of you will be sent to a distant and isolated room and the outputs should be chosen just after receiving the input. In this way, there will be no way for you to communicate after the input is received.

Alice: I see... But before starging to play, do we have some time to discuss, decide our strategy, etc?

Referee: Yes! You have all the time you want to decide a common strategy. We can start whenever you are ready.

Alice and Bob: Nice, we are in!

After the discussion, the referee also gave them a manual with the explicit statements of the rule.

1. After the parties have decided to start, they will be moved to separated and isolated rooms, in a way communication between the parties are not possible.

2. Alice receives a cardxthat can take values 0 or 1. Bob receives a card ythat can take values 0 or 1. Both inputs are going to be chosen in an independent and uniform way. That is, the probability ofx=0 orx =1 is 1/2. Also, the the probability ofy=0 ory=1 is 1/2.

3. Just after receiving the inputs, Alice and Bob should choose their inputs and send them to the referee. A should chose a cardathat can assume values 0 or 1. B should chose a cardbthat can assume values 0 or 1.

4. Winning conditions: In the case where both inputs are one, that is, x = y = 1, the parties win when the outputs are different, that is, a6=b. For all other input combinations,(x=0,y =0),(x=0,y=1), (x=1,y=0), they win when the inputs are the same, that is,a=b. This

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 18 winning condition can be compactly formalised by the mathematical equationa⊕b=xy, where⊕represents the sum modulo two, where we identify 1⊕1=0.

With the rules in hands, Alice and Bob start thinking possible strategies for the game. Alice, that had a good knowledge on classical physics and special relativity (but not much about quantum mechanics) was silent when Bob started the discussion.

Bob: I think we should always output zero, regardless the input we re- ceive. We have four possible combinations of inputs(x = 0,y = 0), (x=_0,y=₁)_,(x =_1,y =₀)_{, and}(x =_1,y =₁), if we always output zero, we will win 3 out of 4 times.

Alice: I agree. . . moreover, I am afraid that this will be our best strategy. . . Moroever, I believe it isphysically impossibletoalwayswin.

Bob: I have conviced myself already that, if the inputs are chosen uniformly randomly and in an independent way, no strategy can win this game with probability greater than 3/4, hence always outputting zero is an optimal strategy. But why physically impossible? In principle we can evencheat, and use some physical system to communicate our inputs and adapt our strategy to the case where both inputs are one.

Alice: Actually, before receiving the inputs, we are very far away. And since we do not have much time to chose our outputs. It follows fromspecial relativitythat this constraint on space and time that no communication is physically possible. Not even a light ray can travel from my room to yours between the time we receive our inputs and choose our outputs.

Hence, special relativity forbids any possibility of communicating, and cheating is not possible.

Bob: Hun. . . I see! So, since we cannot comunicate at all, the best winning strategy is this one that wins 3 out of 4 input combinations, right?

Alice: Yes, and, morover, I canprovethat there is no better strategy. And the maximal winning probability is 3/4.

Alice and Bob have then proved that the probability of winning is bounded by 3/4, that isp(a⊕b=xy)≤3/4. After that, they agreed on the the strategy of always outputing zero, and played the game many times. As predicted, they won roughly 75% of the games played.

Bob, that had some classes on quantum mechanics becomes disturbed by the question “Can we win this game with some probability greater than 3/4 without communicating?”. He wonders if they can outperform this optimal 3/4 winning-probability-strategy by performing measurements on quantum systems. In the next section we will analyse the hypothesis Alice has assumed to prove that the best winning probability is 3/4 and discuss the mathematical formalism behind these kind of games.

1.2 A Framework for Bell Nonlocality

We will now generalise and mathematically formalise the idea presented in the last section. In the game discussed before, Alice and Bob were choosing

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 19 their outputs based on their inputs in a deterministic way. For instance, Alice had decided that if she receives the inputx, she will outputa(x), that is a (de- terministic) function ofx. In a more general perspective, one can also imagine Alice and Bob choosing their outputs in a (not necessarily uniform) probabil- istic way. Also, since they can communicate before receiving the inputs, they can establish a common strategy based on someshared randomness. Sets of probabilities that can be written as probabilistic mixture of deterministic local strategies are namedBell local[3], and are mathematically formalised as:

Definition 1 (Bell Locality). A set of probability distributions represented by p(ab|xy)is Bell local if there exist probabilities distribution pA(.|x,λ), pB(.|y,λ), andπ(.), such that

p(ab|xy) =

∑

λ

π(λ)p_A(a|x,λ)p_B(b|y,λ)_{. (}¹_.¹₎ The shared variableλthat correlates the parties is termed as “local hidden variable”

and the distributions p_A(_.|x,λ), p_B(_.|y,λ)that encodes Alice and Bob strategies as

“response functions”.

We now return to the example of the last section to illustrate the above definition. As discussed, all statistics of the game can be encoded in the probabilitiesp(ab|xy). For this particular scenario of two dichotomic measure- ments per party we need 16 probabilities to describe all possible combinations of inputs and outputs. These probabilities can be represented in many differ- ent ways, for example, as vectors inR¹⁶. For convenience, we here adopt the convention of representing these 16 probabilities in the matrix

P=









p(00|00) p(00|01) p(00|10) p(00|11) p(01|00) p(01|01) p(01|10) p(01|11) p(10|00) p(10|01) p(10|10) p(10|11) p(11|00) p(11|01) p(11|10) p(11|11)









, (1.2)

where the columns represent a probability distribution for the inputsx andy.

We now recall that a local deterministic strategy is completely described by four dichotomic variables,a0,a1,b0, andb1, whereaxrepresents the choice of outputawhen the input isxandbyrepresents the choice of outputbwhen the input isy. Adopting the convention ofLa₀a₁b₀b₁ for the table of probabilities for the case Alice outputsa₀whenx=_0,a₁whenx =1, and Bob outputsb₀ wheny=0,b₁wheny=1, we now list all possible deterministic strategies

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 20

in the probability notation:

L0000= _{1 1 1 1}

0 0 0 0 0 0 0 0 0 0 0 0

, L0001= _{1 0 1 0}

0 1 0 1 0 0 0 0 0 0 0 0

, L0010= _{0 1 0 1}

1 0 1 0 0 0 0 0 0 0 0 0

, L0011= _{0 0 0 0}

1 1 1 1 0 0 0 0 0 0 0 0

; L0100=

_{1 1 0 0}

0 0 0 0 0 0 1 1 0 0 0 0

, L0101= _{1 0 0 0}

0 1 0 0 0 0 1 0 0 0 0 1

, L0110= _{0 1 0 0}

1 0 0 0 0 0 0 1 0 0 1 0

, L0111= _{0 0 0 0}

1 1 0 0 0 0 0 0 0 0 1 1

; L1000=

_{0 0 1 1}

0 0 0 0 1 1 0 0 0 0 0 0

, L1001= _{0 0 1 0}

0 0 0 1 1 0 0 0 0 1 0 0

, L1010= _{0 0 0 1}

0 0 1 0 0 1 0 0 1 0 0 0

, L1011= _{0 0 0 0}

0 0 1 1 0 0 0 0 1 1 0 0

; L1100=

_{0 0 1 1}

0 0 0 0 1 1 0 0 0 0 0 0

, L1101= _{0 0 0 0}

0 0 0 0 1 0 1 0 0 1 0 1

, L1110= _{0 0 0 0}

0 0 0 0 0 1 0 1 1 0 1 0

, L1111= _{0 0 0 0}

0 0 0 0 0 0 0 0 1 1 1 1

.

We can now see that, in this scenario where the parties have dichotomic inputs and outputs, a set of probabilities is Bell local if it can be written as a probabilistic mixture of the above listed deterministic strategiesi.e.,

p=

∑

a₀a₁b₀b₁

π(a0,a1,b0,b1)La₀a₁b₀b₁. (1.3)

1.2.1 Bell inequalities

A fixed number of inputs and outputs defines a bipartite¹Bell scenario. As we saw in the last section, one can easily construct all deterministic local strategies and the set of local distributions is denoted by theconvex hull²of these deterministic vertices. The convex hull of a finite number of vertices is a polytope, that generalises the concepts of polygon and polyhedron toR^dfor some dimensiond.

Although we have defined a polytope as the convex hull of its vertices, every polytope can be equivalently defined as a set of points respecting a finite number of hyperplanes represented by linear inequalities [17]. This result can be understood intuitively by recognizing these inequalities as facets (theR^dgeneralisation of a face of a polyhedron) defining the boundary of the polytope. Please note that if a point does not respect all these a facet-defining inequalities, this point is necessarily outside of the polytope. This inequality representation is then useful for certifying that a given point does not belong to the polytope.

The convex hull of deterministic local strategies of a given Bell scenario is calledlocal polytope, and the inequalities representing its facets as tight³Bell Inequalities. All Bell inequalities can be represented by rational coefficients c_ab|xyand alocal bound Lsuch that

abxy

∑

c_ab|xyp(ab|xy)≤L, (1.4)

1Although Bell scenarios can also be defined in a multipartite case [12], this thesis is focused on bipartite systems.

2The convex hull of a set of pointsxi∈_R^dis the set of all possible convex combinations ofxi. That is, the set of all pointsv∈_R^dthat can be written asv=_∑ipixiwith real coefficientspi≥0 respecting∑ipi=1.

3A non-tight Bell inequality is an inequality that is respected by every point of the local polytope but does not represent any facet.

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 21 and they play a central role in Bell nonlocality, since they provide a simple way to certify that a given set of probabilities cannot be written in the Bell local form (definition 1). The CHSH game inequality p(a⊕b = xy) ≤3/4 presented in section1.1is an example of Bell inequality. When expressed in the notation of equation (1.4), the CHSH game inequality reads as

p(a⊕b=xy) = ¹ 4

p(00|00) +p(11|00) +p(00|01) +p(11|01) (1.5) +p(00|10) +p(11|10) +p(01|11) +p(10|11)≤ ³

4. One can find all Bell inequalities of a given Bell scenario with the help of the Fourier-Motzkin algorithm [18,19]. This algorithm can find the facet representation of a polytope when the vertex one is provided. Also, there exists softwares which implements this facet-finding algorithm [20,21]. The drawback here is that these algorithms are not efficient and may take a very long time to complete their task. Roughly speaking, the time for the algorithm to finish grows exponentially in terms of the dimension of the polytope and becomes intractable for relatively small dimensions. For a formal discussion on the complexity of finding the facets of a polytope in the vertex representation, we suggest [22], and the first discussion on the complexity of finding Bell inequalities is reference [23].

1.2.2 Fine’s Theorem: Another Way to Understand Bell Nonlocality

In1982Artur Fine published an important result on Bell nonlocality. In [24], Fine provided a mathematical proof for an intuitive interpretation of Bell locality. He has shown that a set of distributions is local if and only if there exists a single joint distribution that recovers all probabilities involved in the problem as marginals. This formalises the intuition that “a set of distributions is local when one can, at least in principle, assign consistent probabilities to all possible measurements simultaneously”.

In order to avoid cumbersome notation we present Fine’s theorem⁴ for the particular case of the two dichotomic measurements. We emphasize that, despite this presentation for a particular Bell scenario, Fine’s theorem holds true for any general one.

Theorem1. Let p(ab|xy)be the probability of Alice having the outcome a∈ {0, 1}, Bob having the outcome b∈ {0, 1}when they perform, respectively, the measurements x∈ {0, 1}and y ∈ {0, 1}. The set of distributions p(ab|xy)is local if and only if there exists a joint probability distribution P(a0,a₁,b0,b₁)such that all probabilities p(ab|xy)are recovered as marginals via

p(ab|00) =

∑

a₁b₁

P(a,a₁,b,b₁); p(ab|01) =

∑

a₁b₀

P(a,a₁,b₀,b); p(ab|10) =

∑

a₀b₁

P(a0,a,b,b1); p(ab|11) =

∑

a₀b₀

P(a0,a,b0,b). (1.6)

4Actually, in Fine’s original paper, the theorem is also only provided for this simplest Bell scenario, but his proof methods can be applied to any general Bell scenario.

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 22

1.2.3 The Non-Signalling Paradigm

Another important concept for studying Bell nonlocality is the notion ofnon- signalling. In the Bell scenario, the parties cannot communicate after receiving their inputs, which imposes some constraints in the possible probability distributions the parties can generate. For instance, the output of Alice a cannot depend directly on Bob’s inputy. More formally we have:

Definition2(Non-Signalling). A set of probability distributions represented by p(ab|xy)respects non-signalling when

pA(a|x):=pA(a|x,y) =

∑

b

p(ab|xy) =

∑

b

p(ab|xy⁰) ∀x,y,y⁰ pB(b|y):= pB(b|x,y) =

∑

a

p(ab|xy) =

∑

a

p(ab|x⁰y) ∀y,x,x⁰. (1.7) Perhaps surprisingly, the set of non-signalling distributions is strictly larger then the set of Bell-local ones. For the particular example of the CHSH game, there exits a non-signalling strategy that wins the game with probability one!

For instance, imagine Alice and Bob share a “box” in which they can press input buttons to receive some outputs. Suppose that locally, this box provides uniformly distributed outcomes. That is, the probability of Alice havinga=₀ or a= 1 is 1/2, and similarly to Bob. Since the outputs are independently uniformly random, this box cannot allow signalling. Now, note that we can construct a box that always satisfy the winning condition a⊕b = xy with uniform marginals. In the matrix probability notation defined at equation1.2, we represent the input and output relations of this box as

PR1= ¹ 2









1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0









. (1.8)

The above described box is known as the PR box as a reference to a pion- eering paper by Popescu and Rohrlich that discusses general non-signalling correlations in reference⁵[27].

Since the set of non-signalling correlations is described by finitely many equalities and inequalities, it is also a polytope. Differently from the local polytope, here we can easily obtain the facet representation via the norm- alisation of probabilities, positivity of probabilities, and the non-signalling condition. The same Fourier-Motzkin algorithm [18,19] mentioned before can be used to find the vertex representation when the facets are given. In particular, the vertices of the non-signalling polytope for a scenario with two parties, two inputs per party, and two outputs per input are given by the

5We also mention that these non-signalling correlations that can win the CHSH game with probability one were also previously and independently discovered by two different researchers Rastall [25] and Tsirelson [7,26].

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 23

vertices of the local polytope and eight new ones that take the form of [26] PR₁= ¹

2 _{1 1 1 0}

0 0 0 1 0 0 0 1 1 1 1 0

, PR₂= ¹ 2

_{1 1 0 1}

0 0 1 0 0 0 1 0 1 1 0 1

, PR₃= ¹ 2

_{1 0 1 1}

0 1 0 0 0 1 0 0 1 0 1 1

, PR₄= ¹ 2

_{0 1 1 1}

1 0 0 0 1 0 0 0 0 1 1 1

; PR₅= ¹

2 _{0 0 0 1}

1 1 1 0 1 1 1 0 0 0 0 1

, PR₆= ¹ 2

_{0 0 1 0}

1 1 0 1 1 1 0 1 0 0 1 0

, PR₇= ¹ 2

_{0 1 0 0}

1 0 1 1 1 0 1 1 0 1 0 0

, PR₈= ¹ 2

_{1 0 0 0}

0 1 1 1 0 1 1 1 1 0 0 0

.

1.3 Quantum Nonlocality

In this chapter we introduce some basic notions of quantum mechanics that will be used during this thesis. We will restrict ourselves to the definition of quantum states, quantum measurements, and how they are related to the Bell framework previously discussed. We will not address many important points of quantum theory, for instance, we do not discuss how quantum states evolve in time.

1.3.1 Quantum States

A pured-dimensional⁶quantum state is a normalised⁷vector in ad-dimensional complex linear spaceC^d. As standard in quantum mechanics, we use theDirac bra-ket notation, where vectors are represented by “kets”|ψi ∈C^d, dual vectors by “bras”hφ| ∈_C^∗dand inner products by “brackets”hφ|ψi:= (φ,ψ). Vectors

|ψican be also represented by column matrices and dual-vectorshφ|by line matrices, hence we can understand inner products as standard matrix multi- plication. We also follow the standard convention on quantum information of representingd-dimensional vectors in the orthonormal basis{|ii}^i=d−1_i=0 . Pure quantum states on two-dimensional complex systemsC²are then expressed in the basis{|0i,|1i}, and calledqubits, the short term for quantum bits.

In general, quantum states can be mixed. That is, in addition to pure quantum states, we can also have statistical mixtures of pure states. For example, one may have the pure state |ψi with some probability p and some other pure state|φiwith probability(1−p). Probability mixtures are mathematically formalised by convex combinations and a general description of a quantum state can be conveniently represented by adensity operator. We now define a density operator together with the state postulate of quantum mechanics.

Postulate 1. Every quantum state can be represented by a density operator, i.e., a positive definite linear operator⁸ρ acting onC^dthat has unit trace. In symbolic

6The case of infinite dimension, also known as “continuous variables quantum mechanics”, can be studied in terms of Hilbert spaces. The possibility of having non-converging sequence of vectors poses many technical difficulties in the mathematical formulation of infinite dimen- sional quantum mechanics. In this thesis we avoid these technicalities by sticking to the finite dimensional systems.

7In the standard Euclidean norm.

8We recall that an operator is said to be positive definite if it is self-adjoin (ρ=ρ†) and all its eigenvalues are non-negative.

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 24

mathematical notation we have

ρ≥0;

trρ=_1.

Moreover, every density operator corresponds to a physical quantum state.

In the density operator formalism, pure states |ψi are represented by unidimensional projectors ψ = |ψihψ|. Note that since the pure states |ψi ande^iθ|ψicorrespond to the same density operatorψ=|ψihψ|, pure states with different global phases are equivalent. As we are going to see below, pure quantum states with different global phases lead to the same outcome predictions when submitted to same quantum measurements. It is then a merit of the density matrix formalism to represent equivalent states by the same mathematical object.

1.3.2 Quantum Measurements

Quantum experiments are not only described by states but also measure- ments. Quantum measurements can be seen as the act of extracting classical information from a quantum system. The measurement postulate of quantum mechanics allows us to predict the probability of having a specific output while performing a particular measurement on a quantum state.

Postulate2. Quantum measurements are represented by a set of positive definite operators{Ma}, Ma≥0that sum to identity∑aMa=I. The probability of having a given outcome “a” when measuring{Ma}on the stateρis given by the Born rule

p(a|ρ,{Ma}) =trρMa.

The set of positive operators defining a quantum measurement is also referred as Positive Operator Valued Measure (POVM).

These two postulates already allow us to understand many fundamental aspects of quantum theory. In particular it allows us to discuss the CHSH game discussed in section1.1when the parties use outcomes of measurements performed on quantum states to choose their outputs.

1.3.3 Quantum Entanglement

Composition of quantum systems are mathematically encoded by tensor products. When Alice’s system admits a description on the vector spaceC^dA and Bob’s system onC^dB, the composite system is described on⁹C^dA⊗C^dB. Vectors inC^dA⊗_C^dB are represented by|ψi ⊗ |φi, and as usual in quantum information, whenever it is clear by the context, we use with the shorthand notation|abi:=|ai ⊗ |bi.

9Multipartite quantum states also have a natural representation via the tensor product formalism, but since this thesis only considers bipartite systems, we focus only on the tensor product of two parties.

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 25 Since linear combinations of vectors always lead to valid vectors, the linear spaceC^dA⊗_C^dB admits not only states of the form|0i ⊗ |0ibut

|0i ⊗ |0i+|1i ⊗ |1i

√2 , (1.9)

that can never be written as |ψi ⊗ |φi for any choice of vectors |ψi ∈ C^dA and|φi ∈_C^dB. These bipartite pure quantum states that cannot be written as the tensor product of two vectors are said to beentangled. For general mixed states, quantum entanglement is defined as states that cannot be written as a convex combination of tensor product statesρ_A⊗ρ_B, that is.

Definition3(Quantum Entanglement). A quantum bipartite stateρ_ABacting on the spaceC^dA⊗_C^dB is said to be seperable when there exists quantum statesρ^λ_A,ρ^λ_B and a probability distributionπsuch that

ρAB=

∑

λ

π(λ)ρ^λ_A⊗ρ^λ_B. (1.10) All non-separable states are said to be entangled.

Quantum entanglement plays a central role in quantum information and one main goal of this thesis is to understand better its connection with nonloc- ality.

1.3.4 Bell’s Theorem

We now recall the CHSH game discussed in section1.1. Imagine that Alice and Bob share an entangled two qubit quantum system described by the vector|φ⁺i= ^|00i+|11i√

2 . After receiving the inputs from the referee, they can perform local measurements on their part of the system and choose their outputs based on the outcome of their respective quantum measurements.

We can then imagine that when Alice receives the input x, she performs a measurement described by{A_0|x,A_1|x}and outputsa=0 ora=1 according to the outcome of her measurement. Respectively, when Bob receives the inputy, he performs a measurement described by{B0|y,B1|y}and outputs b=_{0 or}b=1 according to the outcome of his measurement.

Note that the strategy described above does note require any communic- ation after the inputs are received. Also, the probability of Alice and Bob outputtingaandbwhile receiving inputs xandyare now given by Born’s rule, that is,

p(ab|xy) =tr(Aa|x⊗Bb|y

φ⁺ φ⁺

). (1.11) Suppose now that Alice and Bob set

A_0|0=|0ih0|, A_1|0=|1ih1|; A_0|1=|+ih+|, A_1|1=|−ih−|; B0|0=|H+ihH+|, B1|0=|H−ihH−|; B0|1=|h+ihh+|, B1|1=|h−ihh−|,

(1.12) with

|+i:= |0i+|1i

√2 , |−i:= |0i − |1i

√2 , (1.13)

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 26

|H±iand|h±ibeing respectively the positive and negative eigenvectors of¹⁰ H:= −σ_X−σ_Z

√2 , h:= ^σX−σ_Z

√2 . (1.14) Straightforward calculation shows that this quantum strategy provides the winning probability of

p(a⊕b=xy) = ²+√ 2 4 > ³

4. (1.15)

We then see that quantum mechanics does not satisfy the Bell local hypothesis (definition1) and cannot be represented by a local hidden variable theory.

In the beginning of this chapter, Alice and Bob were convinced that it was physically impossible to win the CHSH game with probability greater than 3/4, and we have now shown that simple quantum systems can be used to attain such “impossibility”. This astonishing result is known as Bell theorem, as a reference to Bell’s pioneer paper [3]. Bell’s theorem has deep philosophical implications for quantum mechanics, since it pinpoints one counter-intuitive aspect of the quantum theory and shows that no theory with these natural assumptions of local causality can be used to recover quantum mechanics as a particular case. From a practical perspective, the violation of a Bell inequality has applications in cryptography [28], random number certification [29], and othersdevice independentprotocols. Device independent protocols have their security ensured by some weak assumptions as “the parties cannot communicate between receiving the inputs and outputs” and they can be used even if quantum mechanics predictions do not hold true.

An intense experimental effort was made to reconstruct this quantum strategy for the CHSH game with conditions like the ones presented in section1.1. Some pioneering experiments relied on assumptions that were not included in the standard Bell scenarios, leaving room for someloopholes. For instance, some experiments relied on thefair sampling assumptionto neglect rounds of the game where the measurement apparatus did not fire [30,31], some others did not not ensure that Alice and Bob cannot communicate after the input is received [32,33]. Recently, some research groups managed to address all these loopholes in the same experiment and reported aloophole free Bell inequality violation by performing measurements on quantum systems [34,35,36]. It is then fair to say that we now have strong evidences to believe that “nature” is nonlocal.

1.4 EPR-steering

Suppose Alice and Bob share a two-qubit maximally entangled state|ψ⁻i=

|01i−|10i√

2 . Suppose as well that Alice performs aσ_Zmeasurements, which its POVM elements are given byA0=|0ih0|andA0=|1ih1|. When Alice reads the outcomea=0, her description of Bob’s reduced stateimmediatelychanges from a completely mixed state to|1i. A similar phenomenon happens when

10Hereσ_Z:=|0ih0| − |1ih1|andσ_X=|0ih1|+|1ih0|represent the standardpauli matrices.

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 27 she reads the outcomea=1, in this case she would describe Bob’s state as

|1i.

Please note that, although the thought experiment described above may sound as an astonishing property of entanglement, the same phenomenon can arise even if Alice and Bob share the separable two qubit state ρ_AB =

|01ih01|+|10ih10|

2 . Moreover, this phenomenon of immediately changing the description of a reduced state of a second party after performing local meas- urements has a completely classical analogue. Imagine that a third party named Charlie has a five (Swiss) francs coin that is cut in half. This half coin was cut in a way that it has a “heads side” and a “tails side”. Charlie now put each half coin in a different envelope and mixes these two uniformly. Charlie now sends one of these random envelopes to Alice, and the other one to Bob, who are very distant from each other. If Alice opens her envelope and obtain heads, she now, immediately that Bob has tails,vice versa. We then see that instantaneous change of the description of some other part reduced system is notan exclusive property of entangled systems, and it has a trivial classical analogue. We remark that in the three examples discussed above, there is no change of information between Alice and Bob. That is, the measurement performed by Alice does not send any information to Bob. Moreover, on his side, Bob does not even know that Alice has performed a measurement or not.

We now describe a more general quantum experiment where Alice can choose in performing different measurements. Suppose Alice and Bob share a general quantum stateρABin a way their reduced states are represented by the partial trace¹¹

ρ_A=trBρ_AB, ρ_B=trAρ_AB. (1.16) On her side, Alice has access to a set of measurements{A_a|x}whose inputs are labelled byxand outputs bya. If Alice performs the measurement labelled byx and obtain the outcomea, the description of the reduced state held by Bob ρ_B can be updated with this new information. Quantum mechanics predicts that the upgrade on the description of Bob’s system whenxandais provided is captured by

ρ_B 7→ ρ_a|x= ^trA(A_a|x⊗IρAB)

p(a|x) ^wherep(a|x) =tr(A_a|x⊗Iρ_AB). (1.17) Classical intuition suggests that this experiment could, in principle, be alternatively described in the following way. Bob’s initial reduced stateρ_Bis represented by ρ_B =_∑λπ(λ)ρ_λ, whereλ, a variable that we may not have access to, is distributed byπ(_.)_andρ_λare quantum states. The knowledge of variablesx andaallows us to update the description on the distribution of lambda and it follows from Bayes rule that

p(λ|a,x) = ^p(a|x,λ)π(λ)

p(a|x) ^{. (1.18)}

11The partial trace consists in applying the trace function tr on a single part of the systems. In terms of tensor products it can be defined as trA(ρAB):=I⊗tr(ρAB)where hereIstands for the identity super operator on Alice’s sub-system.

CHAPTER1. QUANTUM MECHANICS AND NONLOCALITY 28 Hence, given the knowledge of x anda, Bob’s reduced state should be up- graded to

ρ_B=

∑

λ

π(λ)ρ_λ 7→ ρ_a|x=

∑

λ

p(a|x,λ)π(λ)

p(a|x) ^{ρλ. (1.19)} For a more compact notation, we now define anassemblageas a collection of the unnormalised states

σ_a|x:=_trA(A_a|x⊗Iρ_AB)_{, (}¹_.²⁰₎ so that we have the identities

ρ_a|x= ^σa|x

p(a|x) ^andp(a|x) =trσ_a|x. (1.21) We say that an assemblage has a classical analogue, or more precisely, it is EPR-unsteerable[1,5], if it admits a representation in terms of an updated distribution on a variableλas in equation (1.19). More formally we have:

Definition4 (EPR-steering). An assemblage{σ_a|x}is EPR-unsteerable if there exists distributionsπ(.), p(.|x,λ), and quantum statesρ_λsuch that

σ_a|x=

∑

λ

π(λ)p(a|x,λ)ρ_λ, ∀a,x. (1.22) At this point, the reader must already suspect that one can prepare steer- able assemblages by performing measurements on quantum states. Indeed, if Alice and Bob share the two qubit entangled stateρ_AB =|φ⁺ihφ⁺|, with

|φ⁺i= ^|00i+|11i√

2 , and Alice performs two dichotomic measurements described by

A_0|0=|0ih0|, A_1|0=|1ih1|

A0|1=|+ih+|, A1|1=|−ih−|, (1.23) the resulting assemblage is described by

σ_0|0= |0ih0|

2 , σ_1|0= |1ih1| 2 ; σ_0|1= |+ih+|

2 , σ_1|1= |−ih−|

2 , (1.24)

and trivially steerable, since pure states cannot be decomposed as other states.

The set of EPR-unsteerable assemblages of a scenario where Alice hasN measurements withd outputs forms a convex set, but not a polytope [37].

An important result on convex analysis states that for every pointx ∈ R^d that is outside a convex setC ⊂ R^dthere exits a a hyperplane separatingx fromC [38]. In the steering context, we name these hyperplanes assteering inequalities[37] and it then follows that all EPR-steerable assemblages violate at least one steering inequality. All steering inequalities can be represented by