### BELL NONLOCALITY

# Bell Nonlocality

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Valerio Scarani

*Centre for Quantum Technologies and Department of Physics, National University of Singapore*

# 3

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*To my parents, Giancarlo and Angela, and to all the others who have taught me to open my eyes in admiration instead of squinting in doubt.*

# **Preface**

*Then I saw that all toil and all skill in work come from a man's envy of his neighbour. This also is vanity and a striving after wind.*

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*Ecclesiastes*, 4:4

*Teacher: may you be eager to make your students understand quickly what has cost you hours of study to see clearly.*

J. Escrivá, *Furrow*, 229

This is a monograph on Bell nonlocality. If you are reading it, you probably have already got an idea of what it is all about—at any rate, all the motivation and introduction have been packed in chapter 1 for the sake of its self-consistency, so I won't repeat them here. Let me just explain what this book tries to be, and what it is certainly not.

This book has been written with the clear awareness that *nonlocality is a phenomenon* i.e., a fact of observation. It is neither speculation, nor a notion mediated by a theory: It can be defined in its own right, and tested in its own right. The fact that quantum theory predicts those observations is a remarkable feat for that theory. The fact that the idea was arrived at by purely foundational thinking, without any application in mind, is an encouragement not to fall into strict pragmatism. But, beyond historical avatars and our current theories, the phenomenon is here to stay—just like apples will keep falling, unaware of Newton and Einstein, indifferent to whether we attribute their trajectory to a force or a curvature.

Because of this, I have opted for presenting the phenomenon from our current perspective: The reader won't find here a presentation of the historical debates or the exegesis of the masters' writings. The book focuses on the concepts and on the mathematical description of nonlocality, both *per se* and in the context of quantum theory. It is structured as follows:

**Part I** contains the core material (although chapters 4 and 5 could be just skimmed through in a first reading).

**Part II** shows the applied side1 of nonlocality: "Device-independent" certification of devices in the context of quantum technologies.

**Part III** extends towards more foundational matters related to nonlocality.

The **Appendices** expand on topics that are important but would have clogged the flow of the main text.

1 Which must not be read as "the dark side": If that were my stance, I would not have devoted so much of my research and a whole part of this book to it.

### **viii** *Preface*

A famous old joke says that Italians write fifty-page pamphlets entitled *Tutto su*, while German multi-volume treatises are called *Einfu*¨*hrung zu*. As an Italian academically formed in Switzerland,2 I have tried to hit somewhere in the middle. In particular:

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Since we are speaking of a review article: This book is not one. I have made a conscious effort to give credit where it is due, notably when attributing priorities; but also, to keep the list of references down to a reasonable length. I have written under the assumption that the internet exists, and that starting from a few relevant references the reader will be quickly able to access related works.

2 I know that Switzerland is not Germany, and besides I was based in Suisse romande—but at least I was North of the Alps for quite a while.

# **Acknowledgment**

Over the years, I have had the chance to interact with many colleagues on the topic of nonlocality (those who have also provided precious feedback on drafts of this book are marked by a∗).

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Two people deserve a very special mention: Antoine Suarez and Nicolas Gisin∗.

Antoine introduced me to this field while I was an undergraduate fascinated by chaos and fractals (already randomness, in a sense). For various reasons I embarked in a Ph.D. on Nuclear Magnetic Resonance (NMR) experiments on magnetic nanostructures:1 While encouraging me to carry it to good completion, Antoine asked me to join him in writing down his idea of before-before experiment. Later he introduced me to Nicolas. After all these years, I am still amazed at his undimmed drive towards unity of knowledge.

Nicolas dared to hire me as a postdoc in quantum information theory, notwithstanding that completely unrelated Ph.D. topic.When hiring me, he left it clear: "*Avec la nonlocalité on ne gagne pas sa vie*". Indeed, I would not be where I am if I had not worked also in cryptography. A few years later, when in a bout of independence I told him that I would resign from his group, he had the class of staying patient, instead of firing me on the spot as I would have deserved. This gave me time to reconsider that silly decision, get back to a positive spirit, and resume work, in time to be part of the pioneering work on deviceindependent certification. Nicolas has just officially retired, but I am sure that neither his brain nor his computer will notice the difference.

The others*...* if I were to tell all the stories, I'd have to write another book, so I'll just list them here in alphabetical order: Antonio Acín∗, Mafalda Almeida, Rotem Arnon-Friedmann∗, Alain Aspect, Jean-Daniel Bancal∗, Jonathan Barrett, Cyril Branciard∗, Gilles Brassard, Cˇ aslav Brukner, Nicolas Brunner∗, Jeffrey Bub, Harry Buhrman, Francesco Buscemi∗, Adán Cabello∗, Daniel Cavalcanti∗, Giulio Chiribella, Andrea Coladangelo∗, Roger Colbeck, Daniel Collins, Michele Dall'Arno, Mauro D'Ariano, Gonzalo de la Torre, Artur Ekert∗, Berge Englert, Chris Fuchs, Otfried Gu¨hne, Michael Hall∗, Lucien Hardy, Paweł Horodecki, Jed Kaniewski∗, Dagomir Kaszlikowski, Adrian Kent, Barbara Kraus, Christian Kurtsiefer, José Latorre, Yeong-Cherng Liang∗, Noah Linden, Alexander Ling, Norbert Lu¨tkenhaus, Lluís Masanes, Serge Massar, Matthew McKague, André Méthot, Michele Mosca, Miguel Navascués∗, Tomasz Paterek, Marcin Pawłowski, Paolo Perinotti, Stefano Pironio∗, Martin Plesch, Sandu Popescu, John Preskill, Renato Renner∗, Grégoire Ribordy, Ana Belén Sainz∗, Nicolas Sangouard,

1 *En passant*, thanks to my Ph.D. supervisor Jean-Philippe Ansermet for being always encouraging to someone who had obviously landed at the wrong place.

### **x** *Acknowledgment*

Ru¨diger Schack, Roman Schmied, Yaoyun Shi, Tony Short, Jamie Sikora, Christoph Simon, Paul Skrzypczyk∗, Rob Spekkens, André Stefanov, Sébastien Tanzilli, Nicholas Teh, Wolfgang Tittel, Philipp Treutlein, Antonios Varvitsiotis∗, Umesh Vazirani, Tamás Vértesi, Thomas Vidick∗, Paolo Villoresi, Stephanie Wehner, Reinhard Werner, Howard Wiseman, Andreas Winter, Michael Wolf, Stefan Wolf, Elie Wolfe, Henry Yuen, Hugo Zbinden, Anton Zeilinger, Marek Z˙ ukowski.

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Since I built my own research group, I have worked with many excellent students and researchers. I proudly keep the full list in the group's webpage. Those who deserve to be mentioned in relation with nonlocality are, again in alphabetical order:2 Charles-Edouard Bardyn, Yu Cai, Wan Cong, Benjamin Goh, Koon Tong Goh, Yunguang Han, Asaph Ho, Melvyn Ho, Yun Zhi Law, Thinh Le, Zhi Xian Lee, Xinhui Li, Charles Lim, Rafael Rabelo, Lana Sheridan, Ernest Tan, Yukun Wang, Xingyao Wu, Tzyh Haur Yang, and Yuqian Zhou. When finishing this book, I decided to put it to the test and proposed a graduate module based on it. Among the students who took it, special thanks to Adrian Nugraha Utama, Ignatius William Primaatmaja, Angeline Shu, and Noah Van Horne for feedback and discussions.

After speaking of the future, a glance to the past. Among my many teachers, I want to single out Alberto Torresani, who taught me philosophy in Liceo Scientifico Argonne, and Fran¸cois Reuse, whose opened my eyes to a proper understanding of quantum theory during my years in Ecole Polytechnique Fédérale de Lausanne (EPFL).

My family and closest friends will hardly understand the title of this book, let alone its content. But without their support, I would not have been able to complete the task.

2 Charles and Rafael have since becomes colleagues, but are mentioned in this list because they started in my group. This is also the reason why Daniel and Jean-Daniel were mentioned previously: When they came to work with me, they had already acquired a great competence in nonlocality.

# **Contents**

### **Part I Classic Bell Nonlocality**

### **xii** *Contents*

# **Part I Classic Bell Nonlocality**

This first part deals with Bell nonlocality *per se*, following a synthetic rather than a historical outline. The first two chapters deal with the definition of Bell nonlocality, the conditions under which it can be demonstrated, and its mathematical formalization. The description of the phenomenon within quantum theory is then introduced, leading to a few general results and a large number of examples.

# **First Encounter with Bell Nonlocality**

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*Misunderstandings of Bell's theorem happen so fast that they violate locality.* R. Munroe, *XKCD*

This chapter serves both as an introduction to the book, and as a self-contained first presentation of Bell nonlocality.

### **1.1 Three Roles for Bell Nonlocality**

Few scientific statements are more radical than one of the core tenets of quantum physics: *There is indeterminacy in nature*. It has accompanied quantum theory since its earliest moments: Only a few months after Heisenberg and Schrödinger independently defined the definitive formalism, Max Born suggested that the laws of the new theory should be seen as intrinsically statistical. This was to become the orthodox view. Sensing the danger, Einstein quickly wrote to Born his conviction that a theory with statistical laws could only be a temporary fix, and that determinism should ultimately be recovered. The debate continued for decades with a few flares, notably the celebrated EPR paper (Einstein Podolsky, and Rosen, 1935) and Bohr's immediate reply, but in an atmosphere of overall indifference among physicists at large. In those years, the excitement about quantum theory was not found in debating its meaning, but in its almost boundless predictive power. It has become commonplace to refer to the attitude of those years by Mermin's dictum "shut up and calculate."

Ultimately, the statistical language became the standard to which generations of physicists conformed out of inertia. If asked for evidence of indeterminacy, still today many would refer to Heisenberg's uncertainty relations, that however can only voice for indeterminacy in quantum theory, not in nature (see Appendix A.1). This is surprising because direct evidence has been compelling since 1964, thanks to the work of John Bell (Bell, 1964). He showed that the possibility of recovering a deterministic model is amenable to experimental falsification, through the observation of a phenomenon that we shall call *Bell nonlocality*. In a first approach, Bell's argument is mathematically simple (see sections 1.3–1.4); because of its importance, it has been submitted to a thorough scrutiny, from which it has emerged unscathed and actually strengthened by more solid foundations (see section 1.5 and chapter 2).

### **4** *First Encounter with Bell Nonlocality*

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In 1964 there was already a huge amount of experimental evidence supporting the validity of quantum theory. Nevertheless, none of those data could be used to check Bell's criterion. Dedicated experiments had to be designed. The work of Alain Aspect and coworkers is credited as the first conclusive evidence of Bell nonlocality (Aspect *et al*. 1982*b*; Aspect *et al*. 1982*a*). The evidence has been steadily growing since then; eventually, three independent experiments reported in 2015 are considered definitive (Hensen *et al*., 2015; Giustina *et al*., 2015; Shalm *et al*., 2015). The main text of this book does not describe experiments; to facilitate reading the experimental literature, a quick guide is provided as Appendix B.

Discovered thanks to quantum theory, indeterminacy has been vindicated as a physical fact, independent of the theory itself. It can be circumvented only at the price of adopting even more radical postures about physics and nature themselves (see section 1.6). *This direct vindication of indeterminism is the original motivation of Bell nonlocality*. For a few decades, it was held to be its sole role too. Those who, for various reasons, were already won to the indeterministic cause had taken note of it and moved on. A series of works that started around 2005 have uncovered a second role: *Bell nonlocality provides the most compelling certification of the correct functioning of some quantum devices*, like those required to perform quantum cryptography and quantum computation. The fabrication of these devices and the development of certification tools based on nonlocality still constitute technical challenges, but we'll have to get there. Far from being an exercise in scientific archaeology, this book contains material that future quantum engineers will have to master—*in nuce* at least, this is a treatise in applied physics.

Finally, as a phenomenon independent of quantum theory, Bell nonlocality is not merely an instrument for a negative task (falsifying determinism): It has a right to citizenship in physics. As its third role,*Bell nonlocality can be used as a principle constraining possible candidates for physical theories*. Barring a few pioneering insights, this approach was also started after the year 2000. It has already contributed several new ideas and notions to the field of foundations of physics but is still very open to future developments.

These three roles of Bell nonlocality—evidence for indeterminism, certification tool for devices, and guideline for foundations—correspond to the three parts into which this book is divided, but pervade the whole text. With them in mind, we can enter the core of the subject.

### **1.2 Introducing Bell Nonlocality**

### **1.2.1 Setting Bell tests: Laboratories and games**

Tests of Bell nonlocality, or *Bell tests* for short, are currently *experimental setups in physics laboratories*. For some years, theorists have rather chosen to present Bell tests as *games* that bear some analogy with TV quizzes, polls, exams, judicial trials, and other familiar situations.1 The game setting is definitely better to bring up the essence of nonlocality,

1 The reader may come back to this list after becoming familiar with Bell test, to find analogies and differences. For instance, in exams, the content of the answer matters, and the verifier will evaluate the

**Figure 1.1** *Sketch of a Bell test for two players (the generalization to more players is straight-forward) in the* laboratory setting *(top) and in the* game setting *(bottom). After having agreed on a process for that round, each player receives an input and has to provide an output; the data of several rounds are then sorted to establish the correlations between the outputs a and b for any pair (x, y) of inputs. In the laboratory setting, we give in grey the usual representation of the process in quantum theory: A quantum state ρ is prepared and some measurements are chosen; which measurement is actually performed in each round is determined by the input. None of this enters the definition of Bell nonlocality. In the game setting, we introduce a verifier V that queries the players Alice and Bob and collects their answers.*

and we shall mostly follow it in this book. Nonetheless, as we shall also see, many important discussions cannot be fully appreciated without going back to the lab. The two settings are sketched and compared in Figure 1.1.

In a Bell test game, the *players*, referred to alphabetically as Alice, Bob, Charlie, etc., are all on the same team. The game consists of many *rounds*. In each round, the players will be separated: Each will receive a query (*input*) and will have to provide an answer (*output*). It is useful to think of a *verifier* distributing the inputs2 and collecting the outputs.

The rules of the game and the list of possible queries are known in advance. The players are allowed to prepare a common *strategy* before the game, which consists in deciding which *process* they will use in each round of the game. We shall also speak of the *resources* that are used in these processes. If the players were allowed to communicate among each other during the game, they would actually not be separated and could easily win any game of this type: The most powerful resources are *signaling* ones. The case is more interesting with *no-signaling resources*. The most elementary example of a nosignaling resource is a list of pre-determined outputs, one for each possible input (that is, the process consists in producing the output by reading the list). It is no-signaling

performance of each player without any concern for correlations. In judicial trials, the players' goal is to provide a consistent version of the story (which may not be the truth), but they don't know in advance the set of questions that they may be asked; etc.

2 In actual experiments, the inputs are usually generated at each player's location by a "random number generator": The image of the verifier allows us to postpone the delicate discussion about randomness and its generation with physical means (subsection 1.5.3).

because, if Alice does something to her list, the other players obviously won't notice anything. In other words, Alice can't send a message to others by manipulating her list. For instance, consider three games, each defined by one of the following rules:

A game based on rule (i) is trivially won by the players agreeing on a fixed common output. A game based on rule number (ii) can be similarly won by agreeing on a predetermined output for each input, provided that the number of inputs is not larger than the number of outputs. If there were more inputs than outputs, the game cannot be won with a list of pre-determined answers. Finally, no strategy based on pre-determined answers can win a game based on rule (iii).

### **1.2.2 The definition of Bell nonlocality**

*Bell locality* means that *the process by which each player generates the output does not take into account the other player's input*. In other words, all correlations between the players' outputs is due to the shared resource, on whose nature no assumption is made: It can be anything, from a list of numbers on a piece of paper to two jointly programmed quantum computers. When Bell locality does not hold we speak of *Bell nonlocality*.

This notion of locality can be formalized as follows. Denote by *λ* the process. It does not need to be deterministic, so we can say that Alice generates *a* by sampling from a probability distribution *Pλ*(*a*|*x*). What is crucial is that this does not take Bob's input *y* into account. Similarly, Bob generates *b* locally by sampling from a probability distribution *Pλ*(*b*|*y*). If this is the case, the statistics observed by the verifier (who is not privy to *λ*) will be described by

$$P(a,b|\mathbf{x},\mathbf{y}) = \int d\lambda \, Q(\lambda) P\_{\lambda}(a|\mathbf{x}) P\_{\lambda}(b|\mathbf{y}) \tag{1.1}$$

where *Q*(*λ*) is the probability distribution that describes the strategy, i.e., how often a specific process *λ* is used. Bell locality is clearly a restriction: Not all conceivable *P*(*a*, *b*|*x*, *y*) can be written in this form. The extreme counterexample is a strategy that wins the game in which each player is supposed to output the other player's input. Also, any winning strategy for the game based on rule (iii) requires one of the players to sample from a distribution that depends on the other player's input.

Statistics will be called *local* if they can be written in the form (1.1), *nonlocal* if they cannot. A *Bell test* is a game whose winning strategy is described by nonlocal statistics.

### **1.2.3 On resources and semantics**

If nonlocal statistics are observed, the verifier knows that the players have shared a *nonlocal resource*. If local statistics are observed, we can't say much about the resource: The players might have shared a potentially nonlocal one but have used it poorly. This sounds like elementary logic, but it triggers two crucial remarks:

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Now, were it not for quantum theory, the definition of Bell nonlocality would sound both uninteresting and uncontroversial: Nonlocal resources would be communication devices. Quantum theory4 however forces us to enlarge, at least in principle, the list of possible nonlocal resources. Let us then consider that the players *share physical systems* in a state that quantum theory describes as *ρAB*, and let's assume that the process that produces the outputs is *performing local measurements* on this state. Specifically, upon receiving her input *x*, Alice performs a measurement on her system, with the output *a* of that measurement is associated to the positive operator *x a*. Bob acts similarly. After several rounds, all played with this process, quantum theory predicts that the statistics collected by the verifier are given by

$$P(a,b|\mathbf{x},\mathbf{y}) = \operatorname{Tr}(\Pi\_a^{\mathbf{x}} \otimes \Pi\_b^{\mathbf{y}} \rho\_{AB}).\tag{1.2}$$

In general, these statistics cannot be cast in the form (1.1): This is the content of *Bell's theorem* (Bell, 1964). Explicit examples will fill this book, but for the time being let us accept that *some shared quantum states are nonlocal resources*. However, it is also well

specific aspects of quantum information theory are summarized in Appendix C.

3 In the same vein, notwithstanding the frequent replacement of "local" with "classical" in the field's jargon, the definition of Bell nonlocality does not rely on a definition of classicality, and even less on assuming the existence of intrinsically classical physical systems. Overall, we shall avoid speaking of classical/quantum systems or phenomena. It is correct to speak of classical theory and quantum theory, because these are well-defined (see Appendix C.1.1 for the essentials). It is also customary to speak of classical/quantum *information* to refer to the resources, insofar as described within each theory, but we won't do it. 4 Familiarity with elementary quantum theory is given for granted in this book; more advanced topics and

### **8** *First Encounter with Bell Nonlocality*

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known5 that shared quantum states *are not communication channels*: By acting only on her system, Alice cannot learn anything about what Bob has done with his—he could have measured it, kept it, discarded it, and Alice does not see any change in her statistics. In this sense, quantum states are *no-signaling resources* just as shared lists of numbers. *Bell nonlocality is interesting and intriguing because it can be demonstrated by sharing no-signaling resources*.

Or can it? Famously (or notoriously), quantum theory does not provide any recipe for the generation of each round's output. Would it be possible that what quantum theory describes as no-signaling resources are actually signaling ones? Einstein dubbed this possibility "spooky action at a distance". As we shall see in section 1.6, it is one possible interpretation. Among those who oppose it, some think that the wording "nonlocality" evokes too closely this unwelcome interpretation. What we called "locality" in subsection 1.2.2, they'd rather call *local realism* or *local causality*. These are elegant expressions with philosophical appeal: They remind us that we are not merely dealing with operations and observations, but with a prejudice in our *Weltanschauung* that has been shattered. However, they are also not exempt from the danger of being over-interpreted.6

With all their potential limitations, the wordings "nonlocality," "local realism," and "local (hidden) variables" have already enjoyed a few decades of tradition and are most probably here to stay. I hope I have said enough to prevent their misuse, and I shall use them freely. As for interpretations, we shall return to them in section 1.6.

### **1.3 My First Bell Test: Clauser-Horne-Shimony-Holt (CHSH)**

To put these general considerations on concrete grounds, we proceed to describe some specific examples of Bell tests. For this introductory chapter, I have chosen to present five classic examples: One in this section and four in the next; several others will be presented later in the book. An elementary proof that each is indeed a Bell test is given, exploiting Fine's theorem (subsection 2.3.3) that allows considering only strategies based on pre-established answers. The relevance of each test for the certification of quantum entanglement is merely stated, leaving all the calculations for chapters 3–5.

5 Even if this should be elementary knowledge, given the centrality of the claim for the content of this book, the explicit proof is given in Appendix C.1.3.

6 It has become commonplace to split the prejudice of "local realism/causality" into two separate prejudices, "locality" and "realism" (or "causality"). To be at peace with the fact of a violation, it would then be enough to abandon either. *Abandoning locality* may legitimately mean signaling: In (1.1), one would have *P*(*a*|*x*, *y*, *λ*), *P*(*b*|*x*, *y*, *λ*), or both; and this modification is indeed sufficient to generate Bell nonlocality. But the meaning of *abandoning realism/causality* is by far less clear (Norsen, 2007; Gisin, 2012). It cannot mean "abandoning determinism": Determinism is not assumed in (1.1), so abandoning it does not generate Bell nonlocality. Neither should it mean "abandoning any connection with reality," reducing physics to unfounded speculations: If we abandon "local realism" it's because we accept the verdict of observation. Probably, "abandoning realism/causality" is a way of saying that only statistics are speakable, see subsection 1.6.3. But then, this alternative is at a different level than signaling: It is not a mechanism, but the statement that no mechanism should be looked for.

It should be obvious that a Bell test requires *at least two players*, otherwise there is no notion of locality. For each player, there must be *at least two possible inputs*: If some players could only receive one query, those inputs would be known to the other players. Finally, for each input, there must be *at least two possible values for the output*. We start with this simplest scenario.

The inputs of Alice are labeled *x* ∈ {0, 1}, her outputs *ax* ∈ {−1, +1} (labeling is of course arbitrary, this choice is convenient for the calculation to come). The inputs of Bob are labeled *y* ∈ {0, 1}, his outputs *by* ∈ {−1, +1}.

The rule of the game prescribes that Alice and Bob should aim at giving the same answer whenever (*x*, *y*) ∈ (0, 0), (0, 1), (1, 0), but opposite answers when (*x*, *y*) = (1, 1). We consider the score

$$S = \langle a\_0 b\_0 \rangle + \langle a\_0 b\_1 \rangle + \langle a\_1 b\_0 \rangle - \langle a\_1 b\_1 \rangle \tag{1.3}$$

where the average is taken over an arbitrarily large number of rounds. The maximal score is obviously *S* = 4.

To prove that this game is a Bell test, we need to find what score can be achieved with local resources. Invoking Fine's theorem, it is enough to see what happens when Alice and Bob have shared a pre-determined quadruple (*a*0, *a*1; *b*0, *b*1) in each round. The existence of these four numbers entails the existence of a well-defined value for the derived quantity *s* = *a*0*b*0 + *a*0*b*1 + *a*1*b*0 − *a*1*b*1. Since the average of a sum is the sum of the averages, *S* = *s* holds. Now, for every quadruple, either *s* = +2 or *s* = −2. This is readily seen by rewriting *s* = *a*0(*b*0 + *b*1) + *a*1(*b*0 − *b*1): Indeed, if *b*0 = *b*1 the second term is zero, if *b*0 = −*b*1 the first term is zero. In Table 1.1 we list explicitly all sixteen possibilities: Eight of them give *s* = +2 and the other eight *s* = −2. Notice how three out of four pairs of inputs contribute in the same way, but the last pair pulls the sum down (or up if it was negative). This observation will become handy later.

In each round, the verifier sees only the pair (*ax*, *by*) corresponding to the inputs (*x*, *y*) he has sent, so he cannot estimate *s*. However, by performing statistics conditional on each pair of inputs, he can estimate the four *axby* and obtain *S*. Now, if *S* = *s* and each instance of *s* can only take the values ± 2, it follows that

$$|\mathbf{S}| \stackrel{L\mathcal{V}}{\leq} 2.\tag{1.4}$$

Suppose now that the verifier finds *S >* 2: He will have to admit that the players were not sharing pre-established quadruples, nor any resource that can be simulated with them in other words, he will have to admit that *they share a nonlocal resource*.

This Bell test is called CHSH from (Clauser, Horne, Shimony, and Holt, 1969). Notice how the mathematical expression of the test is an *inequality*, here Eq. (1.4). The players will convince the verifier that they have a nonlocal resource if they manage to *violate the inequality*. John Bell's original inequality (Bell, 1964) is basically the same as (1.4), but derived under the assumption that one of the *axby* is exactly equal to 1 (Appendix A.4). Since perfect correlations can be predicted but cannot be observed, that inequality

is sufficient to prove that quantum theory predicts nonlocal resources but is untestable in experiments.

The CHSH test is the workhorse of the field, we'll study it in great detail. Let's just mention that the maximal score *S* = 4 can be reached of course by signaling,7 but also with a hypothetical no-signaling resource called a *PR-box* (Popescu and Rohrlich, 1994) that we'll encounter in chapter 9. However, PR-boxes exist in mathematics but don't seem to exist in nature: With quantum entanglement, the maximal score is *S* = 2 √2 ≈ 2.8284.

7 Though rather obvious, here is one possible way in which the players can score *S* = 4 with signaling. Alice and Bob agree on a bit *a* = *b*. Upon being queried, Alice outputs *a* and sends *x* to Bob; Bob outputs (−1)*xyb*. Notice that Alice's answer is pre-determined, but Bob's is not: He has to wait for *x* before producing it. So, the conclusion that both outputs could not have been pre-determined holds.

This value can be achieved by suitable measurements on the maximally entangled state of two qubits8

$$
\langle \left| \Phi^{+} \right\rangle = \frac{1}{\sqrt{2}} \left( \left| 0 \right\rangle \left| 0 \right\rangle + \left| 1 \right\rangle \left| 1 \right\rangle \right), \tag{1.5}
$$

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and in fact, in a sense to be made precise in chapter 7, *only* by that state and those measurements.

### **1.4 Four more Classic Bell Tests**

This section introduces four other examples of Bell tests that should help gaining further familiarity with these notions.

### **1.4.1 Mermin's outreach criterion**

In his effort to explain Bell nonlocality in a simple way, David Mermin (1981) conceived a Bell test that has become popular. There are two players, each with three inputs (*x*, *y* ∈ {1, 2, 3}) and two outputs (*a*, *b* ∈ {+1, −1}). Let's assume that the observation shows that *ai* = *bi* in all cases where the same input was chosen. We are interested in *O* = *x*,*y P(ax* = *by)* = 3 + *x* =*y P(ax* = *by)*. If the outputs are pre-determined, (*a*1, *a*2, *a*3) = (*b*1, *b*2, *b*3) can take eight values, namely (+1, +1, +1), (+1, +1, −1), etc. For (+1, +1, +1) and (−1, −1, −1), one finds *O* = 9; the other six triples give *O* = 5. Thus, certainly *O* ≥ 5 for local resources. Quantum theory predicts that one can go down to *O* = 4.5. In particular, not even with quantum resources one can win perfectly the game based on rule (ii) defined in subsection 1.2.1, because that would correspond to *O* = 3.

The inequality *O* ≥ 5 defines a Bell test only if *ai* = *bi*. If this assumption is removed, the inequality may be violated with LVs. For instance, the choice of pre-determined outputs (*a*1, *a*2, *a*3) = (+1, +1, +1) and (*b*1, *b*2, *b*3) = (−1, −1, −1) gives *O* = 0. This is the same weakness of Bell's original criterion, which had to be transformed into CHSH to become robust. Robust Bell tests for two players, three inputs and two outputs will be discussed in section 4.2.

### **1.4.2 Greenberger-Horne-Zeilinger (GHZ) test**

The original nonlocality test by Daniel Greenberger, Michael Horne, and Anton Zeilinger (1989) considers four players, but the argument can be made for any number

8 The notation |0|0 stand for |0 ⊗ |0. Here and in the rest of the book, the tensor product symbol is usually implicit when writing quantum states, while it is often explicit when writing operators. I find that this is the choice that facilitates the reading.

of players larger than two. Nowadays, we refer to the three-players version as the "GHZ test" without further qualifiers (Mermin, 1990b); this is the one we present here. As in the CHSH case, each player has two inputs and two outputs, those of Charlie being labeled *z* and *c*. Assume now that the verifier observes the following perfect correlations:

$$
\langle a\_0 b\_0 c\_1 \rangle = \langle a\_0 b\_1 c\_0 \rangle = \langle a\_1 b\_0 c\_0 \rangle = +1. \tag{1.6}
$$

Then,

$$
\langle a\_1 b\_1 c\_1 \rangle \stackrel{L\mathcal{V}}{=}+1.\tag{1.7}
$$

Indeed, *axbycz*=+1 means that *axbycz* = +1 in each single round. If the outputs are pre-determined, they must be such that *a*0*b*0*c*1 = *a*0*b*1*c*0 = *a*1*b*0*c*0 = +1 in each round. By multiplying the three conditions and noticing that *a*2 0 = *b*2 0 = *c*2 0 = 1, we get *a*1*b*1*c*1 = +1. In quantum theory, however, one can have the correlations (1.6) alongside

$$
\langle a\_1 b\_1 c\_1 \rangle = -1. \tag{1.8}
$$

This is obtained with suitable measurements on the state

$$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle\langle 0|\langle 0| + |1\rangle\langle 1|\rangle\langle 1|\right),\tag{1.9}$$

and, just as for CHSH, this quantum realization is unique (see chapter 7).

As presented, the GHZ test relies on the observation of perfect correlations or anticorrelations. To cope with unavoidable imperfect situations, its score can be turned into the so-called *Mermin inequality*:

$$M = \langle a\_0 b\_0 c\_1 \rangle + \langle a\_0 b\_1 c\_0 \rangle + \langle a\_1 b\_0 c\_0 \rangle - \langle a\_1 b\_1 c\_1 \rangle \stackrel{LV}{\leq} 2. \tag{1.10}$$

The proof that inequality (1.10) holds is left as Exercise 1.1. Thus, contrary to what happens with CHSH, the maximal score *M* = 4 of the Mermin inequality can in principle be attained in quantum theory. We shall complete the study of this inequality and its generalizations for more parties in section 5.2.

### **1.4.3 Hardy's test**

At this point, one may ask if one can build a Bell test for *two* players on extreme correlations that quantum resources can in principle achieve. There are indeed such examples. The first one was found by Lucien Hardy (1992; 1993). The extreme probabilities that are enforced are:

$$P(a\_0 = +1, b\_0 = +1) = 0\tag{1.11}$$

$$P(a\_0 = -1, b\_1 = +1) = 0\tag{1.12}$$

$$P(a\_1 = +1, b\_0 = -1) = 0.\tag{1.13}$$

The following inferences are then obvious:

By enchaining these three inferences we are led to a fourth one, namely "*b*1 = +1 implies *a*1 = −1," and thus in particular to the prediction

$$P(a\_1 = +1, b\_1 = +1) \stackrel{LV}{=} 0.\tag{1.14}$$

Another way of reaching the same conclusion is suggested in Exercise 1.2.

In subsection 4.4.1, we shall see that in quantum theory one may have the three constraints (1.11)–(1.13) and nonetheless *P*(*a*1 = +1, *b*1 = +1) *>* 0 with a maximal value of approximately 0.11. This looks absurd, since enchaining the bullet pointed three inferences looks innocuous—but it is not: Just as in the derivation of the CHSH inequality, the enchaining assumes that one can speak of both *a*0 and *a*1, and of both *b*0 and *b*1. Rigorously, the first inference should read: If *b* = +1 was found for *y* = 1, we know that *a* = +1 will be found *if the input x* = 0 *is called*; and similarly for the others.

### **1.4.4 The Magic Square**

The rules of the Magic Square test9 (Cabello, 2001*b*; Cabello, 2001*a*; Aravind, 2002) are slightly more complex. The test has three possible inputs per player, *x*, *y* ∈ {1, 2, 3}; the players are asked to output three bits each: *ax* = *(a*1 *x* ,*a*2 *x* ,*a*3 *x )*, *by* = *(b*1 *y* ,*b*2 *y* ,*b*3 *y )*. These outputs should ideally satisfy the following conditions:

$$\prod\_{j} a\_{\mathbf{x}}^{j} = +1, \prod\_{h} b\_{\mathbf{y}}^{h} = -1, \text{and } a\_{\mathbf{x}}^{j=\mathbf{y}} = b\_{\mathbf{y}}^{k=\mathbf{x}}.\tag{1.15}$$

To see that theseconditions are impossible to fulfil perfectly with pre-determined outputs, let us arrange the nine pre-determined bits in a 3 × 3 square. Upon being queried, Alice outputs the *x*-th line of her square, and Bob the *y*-th column of his. The third condition says that the value at the intersection should be the same for every call (*x*, *y*) of the verifier, which means that the squares must be identical. However, the first condition

9 The square made its first appearance as a test for single-player "contextuality" (see Appendix D.3 for this notion) in works of N. David Mermin and Asher Peres. Because of this, it is often called Mermin-Peres Magic Square. Here I cite the subsequent works that introduced it as a two-player nonlocality test.

implies *x*,*j aj x* = +1, the second *y*,*k bk y* = −1. If these are to be enforced, Alice's and Bob's 9-bit squares should differ in at least in one bit—and then, if the verifier calls precisely those inputs, he will see that the third condition fails.

All in all, with pre-determined outputs, Alice and Bob can satisfy the three conditions (1.15) for at most 8/9 of the rounds on average; but there exists a quantum state and measurements that can fulfil them perfectly, as we shall show in subsection 4.4.2.

### **1.5 A Closer Scrutiny: Addressing Loopholes**

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As we have just seen, Bell tests can be described in very elementary terms. But is this not too elementary, especially given the strong conclusions that are reached? Over the years, Bell tests have been submitted to tight scrutiny, searching for flaws, or *loopholes*, in the reasoning or in the implementations.

The four possible loopholes that have been identified turn out to be very different from each other: Some are mere technical fixes (that have been fixed), others border on philosophy and can be closed only under reasonable assumptions (in other words, there is a price to pay if one wants to believe that they are still open). We review them here in this order; their working will be illustrated with the CHSH test.

### **1.5.1 The "memory loophole," or doing proper statistics**

The memory loophole is related to statistics. Basic statistics assumes that rounds of a test are *independent and identically distributed (i.i.d.)*, but this i.i.d. assumption is obviously unwarranted when it comes to such fundamental tests. Would it be possible for the players to give a false positive in a Bell test, i.e., violate a Bell inequality with local resources, by adopting a non-i.i.d. strategy, that is, by choosing the process to be used in round *r* based on all that has happened in the previous rounds? The answer is no.

To appreciate why, consider one round of the CHSH test. The players must choose the pre-established quadruple of values to be used in that round. They can base their choice on whatever piece of information from the past: There will always be one pair of inputs which pulls the sum in the wrong direction (see Table 1.1), and the verifier might have picked precisely that pair; so |*S*| ≤ 2 still holds. This simple argument shows that there is only one way for the players to generate a false positive: Avoid the wrong pair of inputs, either by refusing to answer or by colluding with the verifier. These are, respectively, the fair-sampling loophole and the free-will loophole to be described.

Thus, even if we initially derived our Bell inequality thinking in i.i.d. terms, we have proved that *there is no memory loophole* if the Bell test is infinitely long and the verifier can extract perfect statistics. In a real test, when only finitely many rounds are possible, the Bell test must be phrased as hypothesis testing: How likely is the observed string of outputs assuming that the players are using a local resource? For such likelihood bounds, non-i.i.d. estimators must indeed be used instead of the familiar i.i.d.-based Gaussian standard deviations. We'll get back to this point in section 2.6.

### **1.5.2 The "detection loophole(s)," or the dangers of post-selection**

If in an exam the students were allowed to decline to answer till they are asked a question of their liking, the average score would be certainly increased. The same happens for nonlocality. In the CHSH test played with local strategies, we have seen that only one pair of inputs (*x*, *y*) pulls the value of *S* in the wrong direction: If the players were allowed to decline answering when they receive that specific query, the verifier would never catch them at fault. There is only a subtlety: Neither of the players knows the pair (*x*, *y*), so the decision to decline must be made locally, based on *x* or on *y* alone. A possible strategy in which Alice answers always while Bob is in charge of declining is given in Table 1.2. We refer to Exercise 1.3 for a thorough study of this strategy, and to Appendix B.3 for a more rigorous quantitative approach.

The fix for this loophole is clear: *The verifier must elicit an answer in every round*. This does not sound to be a big deal, but it may be. Let us look at this loophole from the perspective of an honest experimentalist who possesses a very good nonlocal resource (say, a source of photons entangled in polarization) and very accurate measurement devices, but whose detectors have poor efficiency. If she is obliged to produce outputs in every round, in most of the rounds she'll have to produce a dummy output because the detectors won't have fired. Nonlocality is quickly washed down, and likely the observed data will be compatible with a local resource.

The situation is all the more annoying because this loophole has a very conspiratorial character. As John Bell himself stressed and many after him, quantum theory provides

**Table 1.2** *A strategy exploiting the detection loophole for the CHSH test. In each round, Alice and Bob choose one of the eight quadruples of pre-established values that would give s* = +*2 (c.f., Table 1.1). Alice answers always, while Bob declines to answer to the input indicated by brackets, in such a way that the problematic output (boldface) is never produced.*

a very accurate description of such an experiment. It's very hard to believe that this accuracy is just an accident due to detectors being inefficient. Besides, the mechanism of the loophole assumes that a detector's firing depends on the input of the Bell test chosen in each round (to continue with the example, the polarization basis chosen for the measurement). But experimentalists know that the detector's firing depends on internal parameters: With respect to the inputs of the Bell test, the detector is performing a *fair sampling*.

This is why, rather than claiming failure, all the early experiments have reported the observation of nonlocality under the fair-sampling assumption. Most likely, several future experiments will legitimately continue to do so. That being said, it's also important to put to record that nonlocality without the fair sampling assumption has been observed, first in an experiment with entangled ions (Rowe *et al*., 2001), then in several other platforms, including of course the three "loophole-free" experiments of 2015 cited previously.

Other loopholes related to the process of detection have been identified for some specific implementations: We refer the interested reader to the comprehensive review by Larsson (2014). Ultimately, they can all be closed by a rigorous implementation of the rule "one output for every input."

In summary, although they may prove challenging for some platforms, *the detection loopholes can be closed* and are therefore not a threat for the certification of nonlocality.

### **1.5.3 The "free-will loophole," or measurement independence**

Instead of allowing the players to decline answering, the verifier could reveal some information about the inputs he is going to send out in every round. In the CHSH test, it is enough to inform the players about a pair of inputs that won't be used in a given round (in fact, more than enough: See Exercise 1.4).

This possibility sounds as artificial as the previous one, because the verifier has no reason to reveal that information. But there is a crucial difference. In the case of the detection loophole, it is easy to enforce an answer in every round (if the players refuse to comply, the verifier can fill the answer himself and too bad for them). Here, the verifier has to *ensure that no information leaks out to the players*. As frequent reports of leakage and hacking confirm, this is notoriously much more difficult to check, and ultimately impossible to guarantee in an absolute way.

In the laboratory setting, this loophole is open if the preparation of the system to be measured is correlated with the measurements that are going to be performed. This possibility is called *measurement dependence*; though less frequently than hacking, it has also been in the news.10 The ultimate form of measurement dependence is super-determinism; leaving metaphysics for section 1.6, let us assume that measurement independence is possible in principle, and ask: How can one try and enforce it?

10 In 2015 it was discovered that a car manufacturer had programmed some models to detect the specific procedures of an anti-pollution test. The car's emissions would then be lowered in order to pass that test, before returning to the normal, environment-unfriendly settings. As we see, measurement dependence exists; but it is rarely the result of an accident and is usually taken as the evidence of conscious tampering.

For some, the ultimate enforcement of measurement independence would be to choose the inputs using human free will; whence the name of "free-will loophole." I prefer to leave such a delicate notion as free will out of the picture.11 At any rate, all that is required is to choose the inputs with *a process that is very unlikely to be correlated* with the resources shared by the players.12 Some have gone as far as to generate the inputs of a Bell test from fluctuations of radiation coming from very distant stellar objects, i.e., produced by matter that has not been in contact with earthly matter since inflation, if ever (Handsteiner *et al*., 2017). Impressive for a physicist, this choice of process may not be convincing for a technological skeptic: Who guarantees that those telescopes and electronics are really producing stellar randomness? This is why others have argued that the best way to approach the free will loophole is to generate the inputs from the letters of one's favorite book or the Geneva phonebook (Pironio, 2015). Surely there are several ways in which these inputs may not be called random: There is a structure in the text, and the information has been available in the Universe for quite some time—still, in order to refuse the evidence of Bell nonlocality, our skeptics must now believe that the behavior of some physical systems is correlated to the text of a book. If someone does not find this insane, there is little chance that they can be convinced anyway.

In summary: Whether viewed as leakage of information from a verifier, or as hidden correlations among the devices in a laboratory, we are in the presence of a loophole that *can only be closed under reasonable assumptions*. In this book, *we shall always assume that measurement independence holds* till section 11.4, where we shall see that one can relax it partially and still be able to certify Bell nonlocality. Finally, for interpretational matters it is crucial to stress that *measurement independence is compatible with determinism*: It requires that there exist several uncorrelated chains of events, but each chain can be deterministic. In other words, by assuming measurement independence, we are not introducing indeterminism by *fiat*, just as we are not required to accept any modality of human free will in order to certify nonlocality.

### **1.5.4 The "locality loophole," or hidden communication channels**

The last loophole has a different flavor than the previous ones. It does not aim at generating a false positive, but a trivial positive: If the nonlocal resource could be *communication*, one says that the "locality loophole" is open. Closing the loophole would mean to design a Bell test, in such a way as to certify nonlocality while guaranteeing that no communication was happening. This seems to be possible using a physical fact: Information propagates at a speed bounded by that of light in vacuum.

11 I cannot resist referring the reader to an introductory book that reviews the debate on free will, accessible to beginners like us physicists (Griffith, 2013).

12 If the process that is used to choose the inputs is pseudo-random, it is a finite-length algorithm and ultimately the players would be able to learn it (Bendersky *et al*., 2016). I cite this in a footnote because, important as it is conceptually, this possibility makes no difference in practice: Real Bell tests achieve excellent statistical significance in far fewer rounds than would be needed to guess even a moderately short algorithm.

**Figure 1.2** *The space-time configuration in which the locality loophole is closed, with the dotted lines representing the light cones. Information about x may arrive at the location of Bob only after the output b has been produced, and information about y may arrive at the location of Alice only after the output a has been produced. There is no absolute way of determining when the information about the inputs was created, nor when a definite output is produced: The events that are relevant to close the locality loophole can only be defined under reasonable assumptions.*

Recall that the information that needs to be communicated are the inputs. For every player *p*, we denote by *x(p)* their position, by *t (p) i* and *t (p) o* the times at which the input is received, respectively at which the output is produced. The verifier wants to enforce that each player produce their output before any information about the other players' inputs may have arrived at their location: In jargon,*space-like separation* between the event "output" of each player and the events "input" of all the other players. This requirement reads13 |*x(p)* − *x(p )* | *> c(t (p) o* − *t (p ) i )* for all pairs of players (*p*, *p*) (Figure 1.2).

Can one find flaws in this argument? Some may question the physical assumption: Maybe some information can propagate faster than light. We shall discuss this option as interpretation in subsection 1.6.1, then as attempted models in chapter 11. But there is a more subtle possible flaw that went unnoticed for decades: In order to run an argument based on space-like separation, one must also be able to *identify the relevant events*. Alice can say when and where the input was fed into her devices; but she can't say when and where the information about the input was created in the Universe. Similarly, Alice can estimate when an electric current left the detector to convey the information to the computer; but is *that* the time at which the actual result is created, the time of the "collapse"?

13 For notational simplicity, we assume that the relative positions of the players are fixed, which is usually the case in implementations. Also notice that, since the condition of space-like separation is Lorentz-invariant, we did not need to specify in which frame events are parametrized.

In summary, *the locality loophole can be closed for known communication channels* between the players, or by adopting operational definitions of the events, as several experiments did.14 Ruling out any unknown form of communication seems to be impossible: Even if the speed of communication is believed to be bounded, we wouldn't know which are the relevant events.

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### **1.5.5 The unknown loophole: Skepticism**

It should be clear that Bell nonlocality has been scrutinized with great rigor. Some diehard skeptics are not convinced: Every now and then, someone claims to have found the flaw in the argument. These claims are usually based on very convoluted arguments and tend to fall into three categories: Utterly wrong (for instance, the alleged counterexample is just a variation of the detection loophole); exegeses of Bell's papers (whether Einstein, Bell, or anyone else was right or wrong is interesting for the history of science, but science should be judged without reference to their authority); deep discussions on the meaning of probability (some of which may hit home but would apply to every statistical statement and not only to the certification of Bell nonlocality: After all, the philosophy of probability and the cogency of statistical conclusions are still debated).

To be sure, by definition one cannot exhaust the list of possible loopholes, and science should always be open to revision. But at this stage, I strongly believe that the burden of the proof should be on the deniers. If anyone has found the flaw, they should be able to write the corresponding algorithm, take two computers that have been preprogrammed together but do not communicate during the rounds of the game, and exhibit a statistically significant violation of a Bell inequality [see e.g., section 9 of (Gill, 2014)]. In the presence of such evidence, all physicists of the "establishment" will be ready to reconsider the matter.

### **1.6 Experimental Metaphysics?**

There is abundant observational evidence for nonlocality with all the loophole closed – the memory and detection loopholes, indisputably; the free-will and locality loopholes, up to assumptions that go unquestioned in virtually all the rest of science, and have been noticed in this context only because of the strength of the claim. So, we are in the presence of a phenomenon that calls for interpretation.

We may want to start by refocusing on the dilemma. It is the same dilemma for any form of nonlocality, but let us just refer to the GHZ test: Given the outputs of two of

14 The definition of the events is related to electric signals: For the input, when a signal leaves the "random number generator" to reach the measurement device; for the output, when the electric signal leaves the detector to propagate to the computer where the information will be stored. The locality loophole was first addressed in (Aspect, Dalibard, and Roger, 1982*a*), but the first experiment using a more proper random number generator was performed years later in the group of Anton Zeilinger (Weihs *et al*., 1998). A few months earlier, the group of Nicolas Gisin had put the emphasis on the distance rather than on the timing of the random number generation, observing Bell nonlocality between players separated by 10km (Tittel *et al*., 1998).

the players, there is only one possible output for the third—and nevertheless, that output was not predetermined. When was it determined then, and how?

Many positions have been put forward, often overlapping with one's interpretation of quantum theory. For the purpose of this book, I have chosen to classify the options in four groups. This being my own classification, to avoid both canonizations and imprecise attributions I have decided not to insert any citation in what follows. For further reading, one can start with some essays of very different, often opposite flavor published with the occasion of the 50th anniversary of Bell's theorem (Fuchs *et al*., 2014; Maudlin, 2014; Werner, 2014; Wiseman, 2014; Z˙ ukowski and Brukner, 2014). There are also two systematic treatises on the meaning of Bell nonlocality, by Tim Maudlin (2011) and Jeffrey Bub (2015); and two popular books by Anton Zeilinger (2010) and Nicolas Gisin (2014)—and of course, a famous collection of John Bell's reflections (Bell, 2004).

### **1.6.1 Group 1: Nonlocal hidden variables**

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The easiest position to describe is that of those who infer from Bell nonlocality the existence of *nonlocal hidden variables*. With this position, one recovers determinism by staying within a *mechanical paradigm*: That is, one can simulate of how nature processes information in order to produce the outputs.

What would these nonlocal hidden variables be? A form of communication (superluminal or even retrocausal, i.e., propagating to the past), the infinitely rigid quantum ether of Bohmian mechanics, a connection in an unknown dimension *...* Whatever they may be, in our (3 + 1)-dimensional space-time they would appear as "influences" carrying information from one location to the other, which Einstein famously dubbed "spooky action at a distance" in his debates with Bohr.15

Now, these hypothetical influences would carry information, only to tweak it in such a way that it looks no-signaling to us. With a positive wink, Shimony called it "peaceful coexistence with relativity". More negative critics rather highlight the conspiratorial flavor: Why would nature use a signal while hiding its use from us? Both this fine tuning and the relation will relativity will be discussed in detail in chapter 11. In particular, there we shall see a quantitative result: In order to remain "hidden," these influences must propagate at an *infinite* speed in their preferred frame.

### **1.6.2 Group 2: Superdeterminism and its friends**

In the second group, I shall put superdeterminism and stances that (at least in my view) are akin to it.

In subsection 1.5.3, we have seen that Bell nonlocality can be demonstrated as soon as the processes that choose the inputs and those chosen by the players are independent. The strongest way to deny this measurement independence is *superdeterminism*: All the events in the Universe constitute a single, deterministic process. There could be somewhat milder ways of *denying measurement independence*: For instance, worldviews à

15 The first record of this expression seems to date from the 1927 Solvay Conference.

la *The Matrix* in which our Universe is a big simulation, maybe not deterministic in origin (it could be run by aliens using their true free will). Needless to say, the consequences of adopting such a worldview extend far beyond solving the conundrum of Bell nonlocality.

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Another option, directly inspired by quantum theory, are the so-called *many-worlds interpretations* that deny that a definite output is ever singled out. These interpretations say that the reversible dynamics of quantum theory is an accurate depiction of the deepest reality, which we do not perceive with our senses but have discovered with our investigations. What is usually deemed irreversible in an elementary reading of the quantum formalism, namely the act of measurement, is nothing else than getting entangled with the apparatus, and then with the environment, with the hard disk that stores the data, with the consciousness of the players *...* The players will perceive definite outputs, and the verifier will certify Bell nonlocality after many rounds, because that's how the rules are set: In every world, there is indeterminacy and nonlocality. But nature is playing all the options, and this deployment of correlations is fully deterministic.

### **1.6.3 Group 3: Only statistics are speakable, a.k.a. the "orthodox"**

Like the many-worlds interpretations, the third group also asserts the correctness of quantum formalism, but in a very different way. Here, quantum theory is the correct way of computing probabilities in the (only) physical world—and there is nothing else we should talk about. Individual rounds of a Bell test (or of any other experiment: interferometers, Stern-Gerlach *...*) are "unspeakable."

As some of my colleagues like to say, this is "just standard quantum mechanics": Indeed, it is what people identify as the orthodox interpretation. But what should be mentioned, is that it implies a significant epistemological discipline. Physics is generally understood as a representation of nature, a study of the constituents of matter and their dynamics — loosely speaking, it should describe "how nature does it." However, a long list of philosophers may find this stance too naive, and the intrinsically statistical character of quantum theory has won several physicists over to the idea that *a law of nature may rather be a way of organizing our knowledge*. Bayesianism becomes the proper language: Probabilities capture someone's degree of belief. A law of physics does not prescribe the belief itself, which is subjective, but how beliefs should evolve given new information (and it is perfectly acceptable that these updating rules be not subjective).

If someone adopts this approach to physics and knowledge, the "intrinsic indeterminacy" of quantum theory adds only a minor element of discomfort: At some point, agents have to give up the possibility of a more refined description, one leading to stronger beliefs. As for the description of Bell tests, contrary to the many-worlds interpretation, the definite outputs are real and the agent has a special role.

The strength and weakness of this position are both simultaneously evident in the way it deals with the GHZ test: One refuses to explain how the third player manages to give that unique answer, but stresses than an agent should definitely bet on that unique answer, if informed about the other two. This position makes pragmatically correct statements and avoids all the problems. For some it's wisdom, for others escapism.

### **1.6.4 Group 4: Hoping for collapse**

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This last position, in a sense, closes the circle. In this view, the outputs are real (contrary to some in Group 2), they are produced through some intrinsically random process (contrary to Groups 1 and 2) and through Bell nonlocality we are learning about this process that does happen in nature (contrary to Group 3).

The challenge here is to describe the process, usually called "collapse," by which the output of each round is generated. There have been several attempts, but none seems to be fully convincing. Because of Bell nonlocality, any collapse model will have to be nonlocal to describe bipartite or multipartite statistics: The process that generates one player's output must take into account other players' inputs. But it seems inescapable that collapse models exhibit some form of nonlocality for single-player processes too: In a measurement of position, the particle must localize itself somewhere, whence the possibility of finding it elsewhere should fall to zero; if a single photon is sent through a beam-splitter and found in one of the beams, it must become impossible to find it in the other beam.

### **1.6.5 Additional remarks**

I have just tried a simple systematization of the current state of interpretations. This matter is complex and can be approached from several angles. I'll go through a few more viewpoints here.

Let us first address the question of whether Bell nonlocality is *experimental metaphysics* that shapes our *Weltanschauung*. Group 1 would certainly claim so. For Groups 2 and 3, Bell nonlocality is just one of the rules that have been set up and does not play a foundational role (when it comes to Group 2, very few facts can claim to play a foundational role in a deterministic worldview). For Group 4, collapse models have first been studied to explain the appearance of a classical macroscopic world, but Bell nonlocality is something that such models are urged to explain too. At any rate, whatever position one reaches after reflection, Bell nonlocality must have entered that reflection: Nobody brushes it off as irrelevant *a priori*.

Next, I find it interesting to compare these interpretations in terms of *resources and information*. We don't have a recipe to describe how the outputs are generated if we stick only to resources that we can control. Group 4 hopes that this is still possible, at least with a special recipe of collapse. Group 1 favors a mechanistic recipe, at the price of introducing unobserved resources, the nonlocal hidden variables. Groups 2 and 3 stick to the resources that we have: For Group 2, all the outputs are generated according to the rules and it's sheer chance that we end up perceiving one rather than another alternative; for Group 3, how nature generates the outputs is not our business as long as our predictions are correct.

Further, let us consider the issue of *whether quantum theory is complete*, which was the title of the EPR paper. Both Groups 2 and 3 would definitely answer that quantum theory is complete; but we have noticed that they differ in what they call "quantum theory": For the ones it's the kinematics and reversible dynamics, for the others the recipe for computing probabilities. Representatives of Group 4 would like to complete the theory with a model of collapse, probably hinting that collapse has always been a desired feature of quantum theory, a statement with which the others would vehemently disagree. Group 1 advocates for the need of quantum theory to be completed, at least in its ontology (our predictive power may well remain the same).

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*Philosophical labels* are also worth mentioning. Based on observation, I can certify that most of my colleagues would like to be called "realist" in the philosophical sense of believing in the existence and intelligibility of an external reality. Conversely, when the debates get heated, it is frequent to hear insults like "idealist," "pragmatist," or "solipsist" thrown to the representatives of the other camp. One may wonder, for instance, where is the realism in Group 3: They would answer that the laws for updating our beliefs come to us from observation. All in all, this kind of labels should be avoided, also because few of us physicists would be able to pass an exam of philosophy on their exact meaning philosophers themselves may not agree!

Another favorite topic of speculation is *where great figures of the past would stand in this debate*. The usual names that come up are of course Einstein, Bohr, and Bell himself. Assuming that he would stick to his aversion to a dice-playing God and to action-at-adistance, Einstein would either think that we don't have yet enough evidence (something like Group 4) or lean towards determinism at higher levels (Group 2). For Bohr, it is clear that he would despise Group 1; if I'd have to bet, his sympathy would go to Group 3. We know more for Bell. He set out to construct a local hidden variable model, probably thinking that it was possible. When he found it is not, he shifted towards Bohmian mechanics, and when collapse models started to be studied he clearly looked at them with great hope. Where he would stand today, given the evidence that collapse models have not delivered much, is only guesswork.

Surely there is much more to it, and the readers will find further inspiration in reading more or in their own reflections. For the purpose of this book, it is time to put an end to this general introduction and to move on to the formalization of Bell nonlocality.

..........................................................................................

#### EXERCISES

**Exercise 1.1** *Prove that the Mermin inequality (*1.10*) holds indeed for deterministic local variables. Hint: Either c*0 = *c*1*, or c*0 = −*c*1*.*

**Exercise 1.2** *Re-derive Hardy's local variables (LV) prediction (*1.14*) by ticking out from Table 1.1 the quadruples of pre-established values that do not comply with the constraints (1.11)–(1.13).*

**Exercise 1.3** *We consider a modification of the detection loophole strategy for CHSH described in Table 1.2. At every round, Alice and Bob choose one of the eight quadruples listed*

*in the Table with probability* 1 8 *. Then Bob applies that strategy of declining with probability* 1 − *p, whereas with probability p he produces the agreed output to whatever input he receives. The verifier computes S using only the rounds in which both Alice and Bob replied.*

*Prove that the verifier will observe S* = 4 1+*p (hint: What is the fraction of rounds in which Bob replies?). Deduce that this strategy gives a false positive for every p >* 0*.*

**Exercise 1.4** *We consider a false positive for the CHSH test based on the free will loophole.*

# **Formalizing Bell Nonlocality**

*L'universo* [*...*] *é scritto in linguaggio matematico e le lettere sono triangoli, cerchi e altre figure geometriche.*

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*The Universe is written in mathematical language and its letters are triangles, circles and other geometrical figures.*

Galileo, *Il saggiatore*

This chapter introduces the mathematical framework for the study of Bell nonlocality.

### **2.1 Bell Scenarios, Processes, and Behaviors**

Virtually all the theory of Bell nonlocality is a study of *probability distributions*. This means that one is describing Bell tests in the asymptotic limit of infinitely many rounds. The application of this theory to real implementations, with their necessarily finite number of rounds, does not differ essentially from any laboratory use of statistical tools (see some specific remarks in section 2.6).

### **2.1.1 Bell scenarios**

A *Bell scenario* is defined by specifying the number of players, the alphabet of inputs1 that each player can receive, and the alphabet of possible outputs (the latter may be different for each input, but we shall only consider Bell tests in which the alphabet of outputs is the same for each input).

From now until chapter 5, we consider two-player, or bipartite, scenarios. The players are usually called Alice and Bob. The (*MA*, *mA*; *MB*, *mB*) Bell scenario2 is one in which Alice (Bob) has *MA* (*MB*) possible inputs and *mA* (*mB*) possible outputs for every input. In general statements, I shall adopt the notation *x* ∈ X = {1,*...*,*MA*}, *a* ∈ A = {1,*...*,*mA*}, *y* ∈ Y = {1,*...*,*MB*}, *b* ∈ B = {1,*...*,*mB*}. In concrete examples, it may be

1 In the literature the inputs are often called "settings," since in the laboratory they determine which

measurement to perform, i.e., how to set the measurement apparatus. 2 For unknown reasons, the conventional usage in the literature is rather (*MA*, *MB*, *mA*, *mB*). I find it more logical to divide according to player, rather than specifying first the alphabet of the inputs and then that of the outputs.

more convenient to adopt other conventions, for instance {0, 1} or {−1, +1} for binary alphabets, and I shall freely do it when no ambiguity is possible.

### **2.1.2 Processes and strategies**

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For every round, before receiving the inputs, the players agree on the *process* by which they will produce the outputs. At this point, the physical implementation of the process can be virtually anything: Computing a function, casting dice, and tossing coins, using quantum computers, discussing over the phone, or even operating as yet-unknown physical resources. The players obviously have to know what they are doing—but we don't need to: We treat the process as a black box and describe it only in terms of how the outputs are related to the inputs. This relation may be stochastic. Therefore, we identify a process *λ* with the collection of the probabilities *Pλ*(*a*, *b*|*x*, *y*) of obtaining the outputs (*a*, *b*) when the inputs (*x*, *y*) were given:

$$\mathcal{P}\_{\lambda} = \{ P\_{\lambda}(a, b | \mathbf{x}, \mathbf{y}) \mid a \in \mathcal{A}, b \in \mathcal{B}, \mathbf{x} \in \mathcal{X}, \mathbf{y} \in \mathcal{Y} \}. \tag{2.1}$$

The probabilities must be non-negative, i.e.,

$$P\_{\lambda}(a, b | \mathbf{x}, \mathbf{y}) \ge 0 \quad \forall a \in \mathcal{A}, b \in \mathcal{B}, \,\mathbf{x} \in \mathcal{X}, \,\mathbf{y} \in \mathcal{Y},\tag{2.2}$$

and properly normalized for each pair (*x*, *y*), i.e.,

$$\sum\_{a \in \mathcal{A}, b \in \mathcal{B}} P\_{\lambda}(a, b | \mathbf{x}, \boldsymbol{\mathcal{y}}) = 1 \,\,\forall \mathbf{x} \in \mathcal{X}, \,\,\mathbf{y} \in \mathcal{Y}. \tag{2.3}$$

For each pair of inputs (*x*, *y*), one has to give all the *Pλ*(*a*, *b*|*x*, *y*) but one, because of the normalization. The number of real parameters needed to describe a process is therefore *D*gen = *MAMB*(*mAmB* − 1).

A *deterministic process* is one in which the outputs are uniquely determined by the inputs, i.e., *Pλ*(*a*, *b*|*x*, *y*) = *δ*(*a*, *b*)=*F*(*x*, *y*, *λ*) = *δa*=*f* (*x*, *y*, *λ*)*δb*=*g*(*x*, *y*, *λ*). The number of deterministic processes is *D* = *(mAmB)MAMB*: For each pair of inputs, one has to say which pair of outputs is being produced.

We also have to say something about the players' *strategy* over all the rounds of the test:

### **2.1.3 The observed behavior**

The verifier does not know which process is used in each round. He observed coarsegrained statistics that are called the observed behavior, or simply *the behavior*, of the players:

$$\mathcal{P} = \{ P(a, b | \mathbf{x}, \mathcal{y}) \, | \, a \in \mathcal{A}, b \in \mathcal{B}, \mathbf{x} \in \mathcal{X}, \mathbf{y} \in \mathcal{Y} \}, \tag{2.4}$$

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where

$$P(a,b|\mathbf{x},\boldsymbol{\uprho}) = \int d\lambda \, Q(\lambda) P\_{\lambda}(a,b|\mathbf{x},\boldsymbol{\uprho}).\tag{2.5}$$

By construction, the *P*(*a*, *b*|*x*, *y*) are also non-negative and normalized:

$$P(a, b | \mathbf{x}, \mathbf{y}) \ge 0 \,\,\forall a \in \mathcal{A}, b \in \mathcal{B}, \mathbf{x} \in \mathcal{X}, \mathbf{y} \in \mathcal{Y} \tag{2.6}$$

$$\sum\_{a \in \mathcal{A}, b \in \mathcal{B}} P(a, b | \mathbf{x}, \mathbf{y}) = 1 \; \forall \mathbf{x} \in \mathcal{X}, \mathbf{y} \in \mathcal{Y}. \tag{2.7}$$

Also, a generic behavior is specified by giving *D*gen real parameters. The behavior is going to be *deterministic* if and only if the players use a given deterministic process *λd* in every round, i.e., *Q*(*λ*) = *δ*(*λ* − *λd*).

### **2.2 No-Signaling Processes and Behaviors**

### **2.2.1 Definition and motivation**

We introduce the following:

**Definition 2.1** *A process λ is called no-signaling if each player's output statistics do not depend on other player's inputs, that is if*

$$\begin{split} &P\_{\lambda}(a|\mathbf{x},\mathbf{y}) \stackrel{\text{NS}}{=} P\_{\lambda}(a|\mathbf{x},\mathbf{y}') \equiv P\_{\lambda}(a|\mathbf{x}) \text{ for all } a \in \mathcal{A}, \mathbf{x} \in \mathcal{X}, \mathbf{y}, \mathbf{y}' \in \mathcal{Y} \\ &P\_{\lambda}(b|\mathbf{x},\mathbf{y}) \stackrel{\text{NS}}{=} P\_{\lambda}(b|\mathbf{x}',\mathbf{y}) \equiv P\_{\lambda}(b|\mathbf{y}) \text{ for all } b \in \mathcal{B}, \mathbf{y} \in \mathcal{Y}, \mathbf{x}, \mathbf{x}' \in \mathcal{X}. \end{split} \tag{2.8}$$

*Similarly, a behavior is called no-signaling if*

$$\begin{split} P(a|\mathbf{x},\boldsymbol{\uprho}) \stackrel{\text{NS}}{=} P(a|\mathbf{x},\boldsymbol{\uprho}') \equiv P(a|\mathbf{x}) \text{ for all } a \in \mathcal{A}, \mathbf{x} \in \mathcal{X}, \boldsymbol{\uprho}, \boldsymbol{\uprho}' \in \mathcal{Y} \\ P(b|\mathbf{x},\boldsymbol{\uprho}) \stackrel{\text{NS}}{=} P(b|\mathbf{x}',\boldsymbol{\uprho}) \equiv P(b|\boldsymbol{\uprho}) \text{ for all } b \in \mathcal{B}, \boldsymbol{\uprho} \in \mathcal{Y}, \boldsymbol{\uprho}, \mathbf{x}' \in \mathcal{X}. \end{split} \tag{2.9}$$

*These conditions are called the no-signaling constraints.The generalization to multipartite scenarios is straightforward.*

Obviously, if all the P*λ* satisfy the no-signaling condition, so will the behavior P, but the converse is not true (Exercise 2.1).

When encountered for the first time, this important definition may sound convoluted: Wouldn't it be simpler to say that each player's output does not depend on the other players' input? Here lies exactly the subtle distinction between no-signaling and local processes and behaviors. The latter will be formally introduced in the next section, but the reader should recall the qualitative discussion of these issues in subsection 1.2.3.

### **2.2.2 Description of no-signaling statistics**

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Because of the no-signaling constraints, it takes fewer parameters than *D*gen to describe a no-signaling process or behavior. The counting can be done as follows. One can first take the marginals as independent parameters: There are *MA*(*mA* − 1) independent *P*(*a*|*x*), and *MB*(*mB* − 1) independent *P*(*b*|*y*). Consider now any choice of (*x*, *y*): For every fixed *b* = *β*, once the marginal *P*(*a*) is given, one is left with *mA* − 1 independent numbers *P*(*a*, *β*); similarly, for every fixed *a* = *α*, once the marginal *P*(*b*) is given, one is left with *mB* − 1 independent numbers *P*(*α*, *b*). All in all, a no-signaling process can be fully specified by giving

$$D\_{\rm NS} = M\_A(m\_A - 1)M\_B(m\_B - 1) + M\_A(m\_A - 1) + M\_B(m\_B - 1) \tag{2.10}$$

independent real numbers.

In line with this parameter counting, we can introduce the convenient *Collins-Gisin representation* of a no-signaling process or behavior (Collins and Gisin, 2004). We show it for the Bell scenarios *MA* = *MB* = 2 and *mA* = *mB* = 2, 3:

The number of entries of the table are indeed *D*NS = 8, 24. All the probabilities that are not written explicitly can be reconstructed from the given ones: For instance, in the second case, one has *P*(*a* = 3|*x*) = 1 − *P*(*a* = 1|*x*) − *P*(*a* = 2|*x*); *P*(3, *b*|*x*, *y*) = *P*(*b*|*y*) − *P*(1, *b*|*x*, *y*) − *P*(2, *b*|*x*, *y*); etc. The example is easy to generalize to any other Bell scenario: There must be entries for each value of the inputs, while one element of each output alphabet can be skipped since its statistics can be reconstructed from the rest; and the marginals can be taken out precisely because of the no-signaling constraints, that make them independent of the other player's input.

### **2.3 Local Behaviors**

### **2.3.1 Definition and motivation**

In chapter 1 we introduced the idea of locality in a Bell test: *In each round*, the process *λ* must be such that each player's output is generated taking only the same player's input into account. At the level of probability distributions, this translates into the following definition, which is obviously central to this book:

**Definition 2.2** *A process is called local if it is of the form*

$$P\_{\lambda}(a, b | \mathbf{x}, \mathbf{y}) \stackrel{LV}{=} P\_{\lambda}(a | \mathbf{x}) P\_{\lambda}(b | \mathbf{y}), \tag{2.12}$$

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*where we have used the traditional label "Local Variables."*

*A behavior is called local if it can be written as a convex combination of local processes, as in (*1.1*). A behavior that cannot be written that way is called Bell-nonlocal, or simply nonlocal.*

Local processes and behaviors clearly satisfy the no-signaling constraints, thus they can also be described by *D*NS real numbers and cast in the Collins-Gisin representation. But a local process is much more constrained, *ad nauseam*, let's us compare the two definitions in words:

### **2.3.2 Local deterministic processes**

Local processes don't need to be deterministic, but the class of local deterministic processes plays an important role.

**Definition 2.3** *A process* P*LD*,*λ is a local deterministic (LD) process if, for any input, the output is deterministic:*

$$P\_{LD, \lambda}(a|\mathbf{x}) = \delta\_{a = f(\mathbf{x}, \lambda)}, \\ P\_{LD, \lambda}(b|\mathbf{y}) = \delta\_{b = g(\mathbf{y}, \lambda)}. \tag{2.13}$$

An equivalent way of characterizing a LD process consists in just giving the list of outputs for all possible inputs:

$$\lambda\_{LD} \equiv \{a\_1, a\_2, \dots, a\_{M\_A}; b\_1, b\_2, \dots, b\_{M\_B}\} \in \mathcal{A}^{|\mathcal{X}|} \times \mathcal{B}^{|\mathcal{Y}|},\tag{2.14}$$

the link with the previous notation being *ax* = *f* (*x*, *λ*) and *by* = *g*(*y*, *λ*). From this notation, it is obvious that the number of LD processes is

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$$\sharp\_{LD} = m\_A^{M\_A} m\_B^{M\_B}.\tag{2.15}$$

These LD processes are all and the only deterministic processes that satisfy the nosignaling constraints (2.8); all the remaining *D* − *LD* deterministic processes can only be realized with signaling resources (Exercise 2.2).

### **2.3.3 Characterization of local behaviors: Fine's theorem**

By definition of Bell locality, the behavior seen by the verifier is a *local behavior*, if and only if there exist a family of local processes *λ* and a distribution *Q*(*λ*) such that

$$P(a,b|\chi,\chi) \stackrel{LV}{=} \int d\lambda \, Q(\lambda) \, P\_{\lambda}(a|\chi) P\_{\lambda}(b|\chi). \tag{2.16}$$

This expression, which we have already encountered (1.1), is the main mathematical object in the formalization of Bell nonlocality: We'll need to find ways to prove that a given behavior is *not* of this form. As it turns out, this general form is not very useful because the number of processes *λ* and the distribution *Q*(*λ*) are arbitrary. Fortunately, there exist two compact characterizations of local behaviors:

**Theorem 2.1** *These two equivalent results are both due to Arthur Fine (1982):*

*(a) A behavior* P *is local if and only if it is a convex mixture of local deterministic processes:*

$$P(a,b|\boldsymbol{x},\boldsymbol{y}) \stackrel{LV}{=} \sum\_{j=1}^{m\_A^{M\_A}} \sum\_{h=1}^{m\_B^{M\_B}} q\_{jk} \delta\_{a=f(\boldsymbol{x})} \delta\_{b=gk(\boldsymbol{y})} \tag{2.17}$$

*with λ* ≡ *(j, k) and j*,*k qjk* = 1*.*

*(b) Using the notation (2.14): A behavior* P *is local if and only if there exists a joint probability distribution* **P***(a*1,*...aMA* ;*b*1,*...*,*bMB), such that each P(a, b*|*x, y) can be obtained as the marginal*

$$P(a, b | \mathbf{x}, \mathbf{y}) \stackrel{LV}{=} \sum\_{\{a\_{\vec{\jmath}} | j \neq x\}} \sum\_{\{b\_{\vec{k}} | h \neq \mathbf{y}\}} \mathbf{P}(a\_1, \dots, a\_{M\_A}; b\_1, \dots, b\_{M\_B}).\tag{2.18}$$

*Proof* Let us start from statement (a). The "if" direction is obvious, because (2.17) is a special case of (2.16). For the reverse, given *P*(*a*, *b*|*x*, *y*) of the form (2.16), one can compute the cumulative distribution *(a)* = *α*≤*a Pλ(α*|*x)*, where we use the labeling *a* ∈ A = {1,*...*,*mA*} of the outputs (labelings being arbitrary, one can redo the proof for any other choice, at the expense of a more cumbersome notation). Let us then add a new local parameter *μA* ∈ [0, 1] and declare that *a* is chosen according to the following deterministic rule:

$$P\_{\lambda,\mu\_A}(a|\mathbf{x}) = \begin{cases} 1 & \text{if } \Sigma(a-1) \le \mu\_A < \Sigma(a), \\ 0 & \text{otherwise.} \end{cases} \tag{2.19}$$

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If *μA* is drawn with uniform distribution, the original process is recovered:

$$\int\_0^1 d\mu\_A \, P\_{\lambda,\mu\_A}(a|\mathbf{x}) = \int\_{\Sigma(a-1)}^{\Sigma(a)} d\mu\_A = P\_\lambda(a|\mathbf{x}).$$

The same construction can be made on Bob's side. Therefore (2.16) can be rewritten as

$$P(a,b|\varkappa,\mathcal{y}) \stackrel{L\mathcal{V}}{=} \int d\lambda \,\mathcal{Q}(\lambda) \int\_0^1 d\mu\_A \int\_0^1 d\mu\_B \mathcal{P}\_{\lambda,\mu\_A}(a|\varkappa) \mathcal{P}\_{\lambda,\mu\_B}(b|\mathcal{y}) \dots$$

Now, as we said, each *Pλ*,*μA (a*|*x)* is deterministic, so it must be equal to one of the *δa*=*fj(x)*; and similarly, each of the *Pλ*,*μB(b*|*y)* must be equal to one of the *δb*=*gk(y)*. The weights *qjk* are just the integrals of the measure *Q*(*λ*)*dλdμAdμB* over the corresponding sets. This concludes the proof of (a). From here, the proof of statement (b) is straightforward. Indeed, on the one hand, LD processes fix the value of the output for all possible inputs, so (*a*) ⇒ (*b*) for

$$\mathbf{P}\left(a\_1 = f\_{\bar{j}}(1), \dots, a\_{M\_A} = f\_{\bar{j}}(M\_A); b\_1 = \mathbf{g}\_k(1), \dots, b\_{M\_B} = \mathbf{g}\_k(M\_B)\right) \equiv q\_{jh}.$$

On the other hand, any assignment of values to all the *ax*, respectively *by*, defines a valid *f j*(*x*), respectively *gk*(*y*); so (*b*) ⇒ (*a*).

### **2.3.4 Examples of local behaviors**

It is useful to look at a few examples of local behaviors.

• The behavior of *one player* can always be reproduced with LV and thus with a mixture of deterministic strategies. In the case of quantum behaviors, *λ* may contain the description of the quantum state *ρ*: The player can then compute on paper the expected probability distribution for any measurement, then generate the outputs according to that distribution. The fresh reader will find this statement obvious, and such indeed it is. More sophisticated readers may have problems with notion accumulated in their formation: Did not great teachers like Feynman and Schwinger derive the quantum formalism from the single-spin Stern-Gerlach measurement? Isn't it established that there cannot be a LV model for states with negative Wigner function? These concerns are addressed in Appendix D.

### **32** *Formalizing Bell Nonlocality*

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$$P(a\_1, \ldots, a\_M, \underline{b}) = \frac{\prod\_{x=1}^{M\_A} P(a\_x, \underline{b})}{P(\underline{b})^{M\_A - 1}} \tag{2.20}$$

such that all marginals match the behavior. Indeed, by first summing over all Alice's inputs but the desired *ax*, one recovers *P(ax*,*b)*; at which point, by construction the sum over all but one of Bob's inputs yields *P*(*ax*, *by*).

• If each player can produce perfect copies of the information shared to define the process, the resulting behavior will be local. Indeed, each player can produce as many copies as his/her inputs, then apply the procedure to obtain the corresponding output, before the run of the test. This is a situation of pre-determined outputs. Thus, all behaviors that can be obtained with perfectly copiable information can also be obtained with pre-determined outputs (for a small modification, see Exercise 2.3).

3 This is a variation over the criterion set forward in one of the earliest reviews on Bell nonlocality (Werner and Wolf, 2001*b*): If it can be done with ping-pong balls, it's not interesting. 4 Notice that this is stronger than requesting that the bipartite behavior is no-signaling: It's requesting no-

signaling even if Bob could actually obtain the outputs corresponding to all his inputs in each round. In quantum theory, which is formally no-signaling, this would always be the case; an obvious case in which *P(b)* exists is that of Bob using a set of commuting measurements.

### **2.4 The Local Polytope and Bell Inequalities**

In this section, we study the set L of all local behaviors, in short the *local set*. We introduce a geometric characterization which is fundamental for the study of Bell nonlocality. The famous Bell inequalities are the conditions for a behavior P to belong to L.

### **2.4.1 The local set as a polytope**

The first simple property of L is *convexity* (Figure 2.1, left): For all *p* ∈ [0, 1] and for all choice of P*LV*,1 ∈ L and P*LV*,2 ∈ L, the convex sum P = *p*P*LV*,1 + *(*1 − *p)*P*LV*,2 belongs to L too. The proof is straightforward: If we use the representation (2.17) for P*LV*,1 and P*LV*,2, the convex sum P is also of the same form with *qjk* = *pqjk*, 1 + (1 − *p*)*qjk*, 2. Convexity is also an intuitive necessity: One possible way of realizing P is to imagine that in a fraction *p* of the rounds one is sampling from P*LV*,1, and in the other rounds one is sampling from P*LV*,2. Clearly, which behavior is used in each round can be decided in advance.

Because probabilities are bounded, the set L is compact. Now, a compact convex set is the convex hull of its extremal points,5 i.e., those points that cannot be written as a convex sum of other points in the set. *The extremal points of* L *are all the local deterministic behaviors and only those*.

**Figure 2.1** *Top: Geometry of the convexity of the local set: If two local behaviors are represented by points in a probability space, the whole line connecting them is also made of local behaviors. Bottom: Sketch of the local polytope L as a subset of the set of no-signaling behaviors (light grey), both embedded in* R*DNS, with DNS* = *2 for the sake of the illustration (the minimal Bell scenario has DNS* = *8). The local polytope as drawn has four extremal points, one trivial facet and three tight Bell inequalities (see subsection 2.4.3). The dotted lines in the third dimension remind that the no-signaling set is a set of zero measure in the space of all possible behaviors.*

5 This is very intuitive; the formal proof goes under the name of Krein-Milman theorem.

### **34** *Formalizing Bell Nonlocality*

Let us prove this statement: We know from Fine's theorem 2.1 that any P ∈ L can be written as a convex sum of LD behaviors: Thus, only LD behaviors may be extremal. *Ex absurdo*, suppose that a LD behavior can be decomposed as convex combination P*LD*,? = *p*P*LD*,1 + *(*1 − *p)*P*LD*,2 with 0 *< p <* 1 and P*LD*,1 = P*LD*,2. The latter two behaviors must differ for at least one pair of inputs (*x*, *y*): For these inputs, there are (*α*1, *β*1) = (*α*2, *β*2) such that *PLD*, 1(*α*1, *β*1|*x*, *y*) = 1, and *PLD*, 2(*α*2, *β*2|*x*, *y*) = 1. But then *PLD*,?(*α*1, *β*1|*x*, *y*) = *p* and *PLD*,?(*α*2, *β*2|*x*, *y*) = 1 − *p*: So P*LD*,? is not deterministic, contradicting the assumption that it was LD.

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Thus, the local set is a bounded polyhedron or *polytope* embedded in R*DNS* (Figure 2.1, right). Since *LD > DNS*, some extremal points are linearly dependent. The (*DNS* − 1) dimensional hyperplanes that delimit it are called *facets* and are in finite number. To be a facet, a hyperplane must satisfy two properties. First, to ensure the dimensionality, at least *DNS* linearly independent extremal points must lie on the hyperplane (unless the zero vector is chosen as to belongs to the facet, in which case the number is *DNS* − 1). Second, all the extremal points that do not lie on the hyperplane must be found on the same side of it.Mathematically, let *v* ∈ R*DNS* is the vector normal to the facet and oriented outside the polytope: If the equation for the points P of the facet is *v* ·P = *f* , then

$$
\sigma \cdot \mathcal{P} \le f \text{ for all } \mathcal{P} \in \mathcal{L}. \tag{2.21}
$$

It is clear that a polytope is fully determined by listing its extremal points, because this information is sufficient to find all the facets. Alternatively, a polytope is also fully determined by listing its facets: Their one-dimensional intersections are the extremal points. In the case of L, the extremal points are easy to list because they are just the LD behaviors, while finding the facets requires some computation.6

### **2.4.2 Assessing the locality of a behavior**

Assessing *whether a given behavior* P *is local* is the elementary task of Bell nonlocality. Essential in theoretical studies, this assessment is also part of the more complex task of assessing whether a real, finite set of data is compatible with local realism, which will be discussed in section 2.6. It is an instance of a "membership problem," since it is asking whether the behavior belongs to the local polytope L.

We just noticed that it's easy to write down the extremal points P*LD*,*i*, so we are going to formulate the problem as: Find *q* = {*q*1,*...*,*qLD* } such that

$$q\_i \ge 0 \quad \forall i, \qquad \sum\_{i=1}^{\sharp\_{LD}} q\_i = 1, \qquad \mathcal{P} = \sum\_{i=1}^{\sharp\_{LD}} q\_i \mathcal{P}\_{LD,i}. \tag{2.22}$$

6 We shall see in section 9.1 that the opposite is the case for another polytope relevant for nonlocality, the so-called no-signaling polytope: There, the facets are easy to list, and one has to work to find the extremal points.

This is a *feasibility linear program*, one the simplest instances of convex optimization. The standard reference for convex optimization is the book of Boyd and Vandenberghe (2004), and we provide a very elementary introduction as Appendix E. Here it is enough to know that, while it is usually impossible to give a solution in closed form even for a linear program, algorithms are known, and packages implementing them exist for most mathematical softwares. These algorithms generate the so-called *dual program* of the original one (which is then called the *primal*) and search for solutions for both.

For the problem under study, the duality of linear programs reflects the dual characterization of the polytope in terms of extremal points or in terms of facets. Indeed, as we show in Appendix E.2.1, the dual of the program (2.22) is a program that checks whether there is a facet that separates the behavior P from the whole local polytope. The algorithm has therefore two possible outputs: If it finds a solution *q* for the primal, it returns this solution, proving that P ∈ L; otherwise, it returns the equation of a facet that separates P from L, proving that P ∈ L. In either case, the verification of the correctness of the output is straightforward.

Although it is obvious from what precedes, I want to highlight explicitly that *the locality of behaviors can be assessed without any prior knowledge of the facets*—that is, in the terminology that we are going to introduce, of the Bell inequalities. This is going to be important, since listing all the facets is a daunting task for most Bell scenarios, as we shall see in chapters 4 and 5.

### **2.4.3 The notion of Bell inequality**

A criterion that separates *some* nonlocal behavior from *all* LV behaviors is generically called *Bell inequality*. Usually, one considers *linear* Bell inequalities, i.e., criteria of the from

$$I(\mathcal{P}) \equiv \sum\_{a,b,\mathbf{x},\mathbf{y}} v\_{ab\mathbf{x}\mathbf{y}} P(a,b|\mathbf{x},\mathbf{y}) \le I\_L. \tag{2.23}$$

As mentioned, the most natural candidates are the *facets of the local polytope*. The number of facets being finite, it is in principle possible to list them all for any Bell scenario.

Not all the facets of L are candidates for Bell inequalities: All the *MAMBmAmB* conditions *P*(*a*, *b*|*x*, *y*) ≥ 0 define facets,7 called *positivity facets* or *trivial facets*.

The equations (2.23) of the non-trivial facets of the local polytope are called *tight Bell inequalities*. Non-tight Bell inequalities are suboptimal for the mere certification of

7 This is not completely trivial: Since L satisfies also no-signaling, one should check if some of the positivity conditions happen to coincide when the no-signaling constraints (2.9) are enforced. As it turns out, this may happen, but only in uninteresting scenarios: Notably, if *mB* = 1 i.e., the only output for Bob is 0, then the no-signaling condition on the marginal of Alice reads *P*(*a*|*x*) = *P*(*a*, 0|*x*, *y*) for all *y*; in other words, all the conditions *P*(*a*, 0|*x*, *y*) ≥ 0 are the same. Also, one may wonder why there are no facets associated to the other boundary, *P*(*a*, *b*|*x*, *y*) ≤ 1. The reason is that the normalization *a*,*b P(a*,*b*|*x*,*y)* = 1 is always assumed, and consequently the upper bound on each *P*(*a*, *b*|*x*, *y*) is a consequence of the positivity of all the other probabilities.

nonlocality but may have additional interesting features, we shall encounter examples in subsection 4.2.3 and in section 5.3.

### **2.4.4 Multiple versions, liftings, and no-signaling forms of an inequality**

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Every Bell inequality appears in *multiple versions* in probability space, due to the freedom of *relabeling*: If a criterion tests nonlocality, the same criterion in which some of the labels have been changed will also test nonlocality in the same way. The possible relabelings are the following:

• One can relabel the inputs of each player. Specifically, if *π* is a permutation over *MA* elements and *π* is a permutation over *MB* elements, then

$$
\sigma\_{ab\text{xy}} \longrightarrow \sigma\_{ab\text{\pi}(\mathbf{x})\text{\pi}'(\mathbf{y})} \tag{2.24}
$$

defines another version of the same inequality.

• Once the labeling of the inputs is chosen, for each input one is free to relabel the outputs. Specifically, for each *x* we write *πx* a permutation over *mA* elements, and for each *y* we write *π y* a permutation over *mB* elements: Then

$$
\sigma\_{abxy} \longrightarrow \sigma\_{\pi\_x(a)\pi\_y'(b)xy} \tag{2.25}
$$

defines another version of the same inequality.

• The most commonly studied Bell scenarios are such that (*MA*, *mA*) = (*MB*, *mb*). In this case, the multiplicity can be further increased by a factor 2 by permuting the name of the players, i.e.,

$$
\omega\_{abxy} \longrightarrow \upsilon\_{byx} \text{ [if}(M\_A, m\_A) = (M\_B, m\_b)].\tag{2.26}
$$

This counting shows that there can be as many as *MA*!*MB*!*(mA*!*)MA (mB*!*)MB* versions of a given inequality, or even twice as many if the players can be exchanged. However, several relabelings may lead to the same version, so the multiplicity of an inequality is usually smaller. The multiplicity must be taken into account: For P to be local, one must check that I*(*P*)* ≤ I*L* holds for *all versions* of all the Bell inequalities in the scenario. This may be bothersome on paper, but recall that a simple linear program does it without the need to give any Bell inequality a priori.

Besides, when moving from a Bell scenario to a larger one (more players and/or more inputs and/or more outputs), all the inequalities of the smaller scenarios are inherited in several versions called *liftings*. As an example, suppose that *vabxy*|*a*, *b*, *x*, *y* ∈ {0,1} defines a Bell inequality in the (2, 2; 2, 2) scenario: If we move to the (3, 2; 2, 2) Bell scenario, we can already anticipate three Bell inequalities, defined by *vabxy*|*x* ∈ {0,1},*a*,*b*,*y* ∈ {0,1} , *vabxy*|*x* ∈ {0,2},*a*,*b*,*y* ∈ {0,1} , and *vabxy*|*x* ∈ {1,2},*a*,*b*,*y* ∈ {0,1} —each of course with its multiplicity due to relabelings. If the original inequality is a facet of the local polytope, the liftings will also be facets of the local polytope for the larger scenario.

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Finally, we must mention that, given a version of an inequality, one can transform the *vabxy* almost beyond recognition by exploiting the normalization of probabilities and even more the no-signaling constraints (2.9). Just to give an idea, consider *a* ∈ {0, 1}: With normalization, one can write 1 = *P*(*a* = 0|*x*, *y*) + *P*(*a* = 1|*x*, *y*), and using nosignaling one can even write 1 = *P*(*a* = 0|*x*, *y*) + *P*(*a* = 1|*x*, *y* ) with *y* = *y*. We shall refer to different *forms* of an inequality, and we shall see several examples, starting from the next section.

Ultimately, all these are symmetries that can be elegantly and efficiently tackled with tools of group theory (Rosset, Bancal, and Gisin, 2014). In this book, we shall deal with these issues in a rather pedestrian way, limiting ourselves to specific examples and occasional warnings.

## **2.5 The CHSH Scenario**

The simplest Bell scenario is (2, 2; 2, 2), usually known as *CHSH scenario* from the work of (Clauser *et al*., 1969) that we have already encountered in section 1.3.

### **2.5.1 The local polytope and the CHSH inequality**

In this scenario, the local polytope is embedded in R8 and has *LD* = 16 extremal points, each being the product of one deterministic process for Alice [(*a*0, *a*1) = (0, 0), (0, 1), (1, 0), or (1, 1)] with one for Bob [same lists for (*b*0, *b*1)]. We have already listed these points, with a different labeling, in Table 1.1. In the Collins-Gisin representation, the extremal points read

$$\mathcal{P}\_{(0,0)(0,0)} = \overbrace{\frac{\begin{array}{c} \begin{array}{|c|c|c|} \hline 1 & 1 & 1\\ \hline 1 & 1 & 1 \end{array} \\ \hline \end{array}}\_{} \mathcal{P}\_{(0,0)(0,1)} = \overbrace{\frac{\begin{array}{|c|c|c|} \hline 1 & 1 & 1\\ \hline 1 & 1 & 1\\ \hline \end{array} \end{array}}\_{} \dots \mathcal{P}\_{(1,1)(1,1)} = \overbrace{\frac{\begin{array}{|c|c|} \hline 0 & 0 & 0\\ \hline 0 & 0 & 0\\ \hline 0 & 0 & 0 \end{array}}\_{}}\_{} \underbrace{\begin{array}{|c|c|c|} \hline 0 & 0 & 0\\ \hline 0 & 0 & 0\\ \hline \end{array}}\_{(2.27)} \tag{2.27}$$

where the completion of the list is left as Exercise 2.4. Having this list, the local polytope is fully determined, and one can set out to find the facets. Even in this simplest of cases, the analytical derivation is tedious, we give it in Appendix G.1 for a meaningful subpolytope. The complete polytope has 24 facets (Froissart, 1981; Fine, 1982; Collins and Gisin, 2004): The expected 16 positivity facets and 8 facets defined by

$$I^{(1)} = E\_{00} + E\_{01} + E\_{10} - E\_{11} \le 2$$

$$I^{(2)} = E\_{00} + E\_{01} - E\_{10} + E\_{11} \le 2$$

$$I^{(3)} = E\_{00} - E\_{01} + E\_{10} + E\_{11} \le 2$$

$$I^{(4)} = -E\_{00} + E\_{01} + E\_{10} + E\_{11} \le 2$$

$$I^{(5)} = -E\_{00} + E\_{01} + E\_{10} + E\_{11} \ge -2$$

$$I^{(6)} = E\_{00} - E\_{01} + E\_{10} + E\_{11} \ge -2$$

$$I^{(7)} = E\_{00} + E\_{01} - E\_{10} + E\_{11} \ge -2$$

$$I^{(8)} = E\_{00} + E\_{01} + E\_{10} - E\_{11} \ge -2$$

where

$$E\_{\rm xy} = P(a = b | \mathbf{x}, \mathbf{y}) - P(a \neq b | \mathbf{x}, \mathbf{y}). \tag{2.28}$$

It is manifest that these are just versions of the same inequality up to relabelings: *I* (2) is obtained from *I* (1) by permuting the labels of Bob's inputs, *I* (3) is obtained from *I* (2) by permuting the roles of Alice and Bob (or by permuting the labels of both Alice's and Bob's inputs), etc. Due to the symmetry of the expression, we have only 8 instead of the potential 2*MA*!*MB*!*(mA*!*)MA (mB*!*)MB* = 128 versions. Thus, up to relabelings, *there is only one tight inequality in the CHSH scenario, namely the CHSH inequality*

$$S = E\_{00} + E\_{01} + E\_{10} - E\_{11} \le 2.\tag{2.29}$$

This is the same CHSH expression (1.3) that we derived in section 1.3. There, we took advantage of the smart labeling *a*, *b* ∈ {−1,+1}to write *axby* = *a*,*b abP(a*,*b*|*x*,*y)* = *Exy*. The current derivation, with *Exy* defined by (2.28), highlights that any labeling of the outputs is allowed, as indeed it should be.

### **2.5.2 The CH form of the CHSH inequality**

Among the infinitely many forms of the CHSH inequality equivalent under no-signaling, the most frequently encountered one is

$$S\_{CH} = \sum\_{\mathbf{x}, \mathbf{y} = \mathbf{0}}^{1} (-1)^{\mathbf{x} \mathbf{y}} P(\mathbf{0}, \mathbf{0} | \mathbf{x}, \mathbf{y}) - P(a = \mathbf{0} | \mathbf{x} = \mathbf{0}) - P(b = \mathbf{0} | \mathbf{y} = \mathbf{0}) \le \mathbf{0},\tag{2.30}$$

called *CH form* because it was introduced by Clauser and Horne (1974). To prove the equivalence, one can plug into (2.29) the expression

$$E\_{xy} = 4P(0,0|\mathbf{x},y) - 2P(a=0|\mathbf{x}) - 2P(b=0|y) + 1,\tag{2.31}$$

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that follows from the relations

$$\begin{aligned} P(0,1|\boldsymbol{x},\boldsymbol{y}) &= P(\boldsymbol{a}=\boldsymbol{0}|\boldsymbol{x}) - P(\boldsymbol{0},\boldsymbol{0}|\boldsymbol{x},\boldsymbol{y}), \\ P(1,0|\boldsymbol{x},\boldsymbol{y}) &= P(\boldsymbol{b}=\boldsymbol{0}|\boldsymbol{y}) - P(\boldsymbol{0},\boldsymbol{0}|\boldsymbol{x},\boldsymbol{y}), \\ P(1,1|\boldsymbol{x},\boldsymbol{y}) &= 1 - P(\boldsymbol{0},\boldsymbol{0}|\boldsymbol{x},\boldsymbol{y}) - P(\boldsymbol{0},1|\boldsymbol{x},\boldsymbol{y}) - P(1,\boldsymbol{0}|\boldsymbol{x},\boldsymbol{y}) \\ &= 1 - P(\boldsymbol{a}=\boldsymbol{0}|\boldsymbol{x}) - P(\boldsymbol{b}=\boldsymbol{0}|\boldsymbol{y}) + P(\boldsymbol{0},\boldsymbol{0}|\boldsymbol{x},\boldsymbol{y}). \end{aligned}$$

The CH expression contains exactly the terms that appear in the Collins-Gisin representation. If the behavior P is written in that representation, one can conveniently write the Bell expression as a term-by-term multiplication of tables:

$$\mathcal{S}\_{CH} = \mathcal{Z}\_{CH} \cdot \mathcal{P}, \quad \text{with} \quad \mathcal{Z}\_{CH} = \begin{array}{c||c|c|c} & -1 & 0 & \\ \hline \hline -1 & 1 & 1 \\ \hline 0 & 1 & -1 \\ \hline \end{array} . \tag{2.32}$$

By the further replacements *P*(*a* = 0|*x* = 0) − *P*(0, 0|0, 1) = *P*(0, 1|0, 1) and *P*(*b* = 0|*y* = 0) − *P*(0, 0|1, 0) = *P*(1, 0|1, 0), *SCH* can be rewritten in the *Eberhard form* (Eberhard, 1993)

$$S\_{\rm Ebe} = P(0,0|0,0) - P(0,1|0,1) - P(1,0|1,0) - P(0,0|1,1) \le 0. \tag{2.33}$$

The numerical values of the three expressions are related through

$$\mathbf{S}\_{\rm CH} = \mathbf{S}\_{\rm Ebc} = \frac{1}{4}\mathbf{S} - \frac{1}{2}\mathbf{.}\tag{2.34}$$

A historical remark must be made at this point. Clauser and Horne presented (2.30) as a new inequality, which is perfectly understandable for those early days. Unfortunately, before the notions of no-signaling and of the local polytope were fully formalized, the belief that CHSH and CH are inequivalent was reinforced by a misunderstanding in the approach to the detection loophole;8 as received knowledge, it has persisted in the literature till very recently.

8 The first studies of the detection loophole addressed it correctly with the CH form, but somehow failed to see how to address it with the CHSH form. As a result, the conviction spread that the latter is simply unsuited for the task. In Appendix B.3 we show how to approach the detection loophole both with the CHSH and the CH form, and obtain the same result.

### **2.6 Proper Statistics: Beyond i.i.d. and Finite Samples**

In this last section, we prove that the local polytope remains unchanged if the underlying strategy is not i.i.d., then discuss some subtleties that one should take into account for a proper statistical assessment of a Bell test.

### **2.6.1 Non-i.i.d. behaviors and the local polytope**

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In this chapter, we have started from a definition of behaviors (2.5) that assumes an i.i.d. strategy. We have built the local polytope from the corresponding definition of i.i.d. local behaviors. The i.i.d. assumption must be removed: As mentioned in subsection 1.5.1, if the players could pass a Bell test by just adopting a local non-i.i.d. strategy, Bell nonlocality wouldn't be worth much discussion.9

For behaviors that describe an asymptotically large number of rounds, we are going to prove that the definition of the local polytope does not change if one relaxes the i.i.d. assumption. For simplicity of notation, we consider processes *λ* that correlate *two* consecutive rounds, but the proof extends easily to any number of rounds. If two rounds are known to be correlated, the proper description of the behavior is10 *P*(*a*1, *a*2, *b*1, *b*2|*x*1, *x*2, *y*1, *y*2). Given this, the single-round statistics can be computed through

$$P(a,b,\mathbf{x},\boldsymbol{\uprho}) = \frac{1}{2} \left[ \sum\_{a\_2,b\_2,\mathbf{x}\_2,\mathbf{y}\_2} P(a,a\_2,b,b\_2 | \mathbf{x},\mathbf{x}\_2,\mathbf{y},\boldsymbol{\uprho}\_2) P(\mathbf{x},\mathbf{x}\_2,\mathbf{y},\mathbf{y}\_2) \right]$$

$$+ \sum\_{a\_1,b\_1,\mathbf{x}\_1,\mathbf{y}\_1} P(a\_1,a,b\_1,b | \mathbf{x}\_1,\mathbf{x},\mathbf{y}\_1,\boldsymbol{\uprho}) P(\mathbf{x}\_1,\mathbf{x},\mathbf{y}\_1,\mathbf{y})$$

where the factor 1 2 comes from the normalization *a*,*b*,*x*,*y P (a*,*b*,*x*,*y)* = 1. Thus, the effective single-round behavior is

$$P'(a,b|\mathbf{x},\boldsymbol{\uprho}) = P'(a,b,\mathbf{x},\boldsymbol{\uprho})/P'(\mathbf{x},\boldsymbol{\uprho}),\tag{2.35}$$

where*P (x*,*y)* = 1 2 [ *x*2,*y*2 *P(x*,*x*2,*y*,*y*2*)* + *x*1,*y*1 *P(x*1*x*,*y*1,*y)*]. Now, suppose that the two-round behavior *P*(*a*1, *a*2, *b*1, *b*2|*x*1, *x*2, *y*1, *y*2) is *local*: By Fine's theorem, it is a convex mixture of local deterministic two-round behaviors. But, any LD two-round behavior is the product of two LD single-round behaviors (the fact that the two rounds may not be independent simply defines the rule with which each single-round behavior is chosen). Thus *P* (*a*, *b*|*x*, *y*) is a convex mixture of LD single-round behaviors, which

9 Still, it took almost forty years after Bell's paper for this matter to be properly addressed by Richard Gill [as early as 2001, see references in (Gill, 2014)] and (Barrett *et al*., 2002).

10 It should be clear that, in this subsection, *aj* does not mean *ax*=*j*, but the output of Alice in the *j*-th round.

means it's local. *A contrario*, if *P* (*a*, *b*|*x*, *y*) were nonlocal, *P*(*a*1, *a*2, *b*1, *b*2|*x*1, *x*2, *y*1, *y*2) must have been nonlocal too.

Notice that the proof does not requires the rounds to be sequential: The result holds even in the case of parallel queries, where the verifier sends all the inputs to the players and elicit all the outputs. In fact, the critical assumption is measurement independence, that we assume systematically till the last chapter (and we'll see in subsection 11.4.3 that, in the presence of measurement dependence, a local two-round behavior may lead to a nonlocal effective single-round behavior).

### **2.6.2 Certifying nonlocality on real data**

Asymptotic statements like the previous one provide a simple working ground for theorists. But real data always come in finite amount, so one has to discuss fluctuations. Among pioneering work, one must cite that of Richard Gill, synthetized in (Gill, 2014), and the first dismissal of the memory loophole by (Barrett *et al*., 2002). Several improvements were reported around the time of the loophole-free Bell tests, we refer to (Zhang *et al*., 2013; Elkouss and Wehner 2016) for a first introduction. These tools are still being refined at the moment of writing: For instance, a Bayesian approach under the i.i.d. assumption has been provided in (Gu *et al*., 2019).

For the level of this book, it is worth pointing out one peculiarity: In the case of nonlocality, *unprocessed raw data fail to provide a point estimator*. The reason is that the local polytope is by construction a subset of the set of no-signaling behaviors, but the

**Figure 2.2** *If observed frequencies are taken as estimators of probabilities, the estimated behavior does not satisfy the no-signaling constraint rigorously. Bell inequalities are defined only in the no-signaling set (NS), because so is the local polytope (L): Thus, plugging frequencies into one's favorite form of a Bell inequality is, strictly speaking, ambiguous. This is illustrated in the figure: The estimated signaling behavior P (respectively, P) is manifestly a fluctuation of a nonlocal (respectively, local) behavior. However, if the inequality is written as B, P does not violate it while P does.*

### **42** *Formalizing Bell Nonlocality*

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estimator of the behavior obtained from frequencies over a finite sample will deviate from the strict no-signaling condition. Plugging such an estimator into the linear program (2.22) will always return a negative result. It's even worse if one plugs the signaling estimator into their favorite form of a Bell inequality11 (subsection 2.4.4): Given a signaling behavior and a Bell inequality as facet of the local polytope, one can always find both a form of the inequality that is violated by the behavior and a form that is not (Figure 2.2). If one wants a point estimator, one first has to project the estimator obtained from the frequencies into the no-signaling set (Lin *et al*., 2018). The error estimator for this procedure is not known at the moment of writing.

..........................................................................................

### EXERCISES

**Exercise 2.1** *Construct an example of a no-signaling behavior as a coarse-graining of signaling processes.*

**Exercise 2.2** *Prove that, in any Bell scenario, deterministic behaviors are either local deterministic or signaling.*

**Exercise 2.3** *Alice and Bob share some resource prior to the Bell test. Suppose that Alice's information can be perfectly copied (while we make no such assumption on Bob's). Prove that the resulting behavior is local.*

### **Exercise 2.4**

11 Unfortunately a large number of experimental papers have done exactly that over the years. I do not doubt that the data could have supported nonlocality (maybe up to loopholes), but the claim is not strictly substantiated.

# **Bell Nonlocality in Quantum Theory**

*To avoid entanglements and interferences had long been one of her first principles.*

C.S. Lewis, *That hideous strength*

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While Bell nonlocality is defined and formalized without reference to quantum theory, it is this theory that describes accurately the known *no-signaling nonlocal resources*. This chapter introduces the study of nonlocality with the quantum formalism.

### **3.1 Quantum Behaviors**

### **3.1.1 Quantum and nonlocality: A preamble**

In subsection 1.2.3 we made the point that Bell nonlocality is intriguing because it can be demonstrated with some no-signaling resources—or more precisely, resources that are accurately described by a formalism that implies no-signaling: Quantum theory.

In order not to disrupt the flow of the study of nonlocality, I won't pause in the main text to introduce this theory. Many readers will likely be very familiar with it. Elementary knowledge at the undergraduate level is assumed, while the more advanced notions are summarily introduced in Appendix C with references for further study.

Quantum theory provides a further very important insight. Nonlocality is not associated to specific physical degrees of freedom (the charge of the electron, the hyperfine constant, the frequency of a light mode *...*). What matters is the possibility of creating and measuring suitable states of composite systems, described as *entangled* by the theory. As long as one can manipulate those states, any physical degrees of freedom goes.

### **3.1.2 Definitions**

In quantum theory, the shared resource is described as a *quantum state* shared among the players. A behavior P is obtained by each player performing a local measurement on their subsystem.

The joint system of Alice and Bob is described by the tensor product Hilbert space H*A* ⊗ H*B*. A state *ρ* is described by a positive Hermitian operator of trace 1. A *mA*-output measurement on Alice's system is described by M*x A* = {*x a*|*a* ∈ A}, where the *x a* are

positive Hermitian operators on H*A* satisfying- *a x a* = I. A *mB*-output measurement on Bob's system is described by M*y B* = {*y b*|*b* ∈ B}, where the *y b* are analogously defined. Notice that the measurements do not need to be projective: They can be generalized measurements (a.k.a. positive-operator valued measures, POVMs).

A behavior P is called *quantum behavior* if there exist a state *ρ* and families of measurements {M*x A*|*x* ∈ X } and {M*y B*|*y* ∈ Y}, such that

$$P(a,b|\mathbf{x},\mathbf{y}) = \operatorname{Tr}(\rho \Pi\_a^{\mathbf{x}} \otimes \Pi\_b^{\mathbf{y}}).\tag{3.1}$$

In chapter 2, from the definition of local behavior (2.16) we derived a compact characterization of the set of all such behaviors, namely the local polytope. The discussion of the set of all quantum behaviors is more complex. Although it could fit here, I will postpone it to chapter 6: I prefer the reader to first become familiar with several *examples* of nonlocality in quantum theory.

### **3.1.3 Relations between states, measurements, and behaviors**

Bell nonlocality is defined on behaviors, while quantum theory is phrased in terms of states and measurements. Here we spell out some evident but important consequences of this difference.

The *states* are the shared resources. In any Bell scenario, given a state, infinitely many behaviors are possible because of the freedom of choosing the measurements. We can introduce here the notion of *state-behavior* as the collection of the statistics of all possible local measurements on *ρ*:

$$\mathcal{P}\_{\underline{Q}}(\rho) = \left\{ P(a, b | \vec{a}, \vec{b}) = \text{Tr}(\rho \Pi\_a(\vec{a}) \otimes \Pi\_b(\vec{b})) \, \middle| \, a \in \mathcal{A}, b \in \mathcal{B}, \forall \vec{a}, \vec{b} \right\} \tag{3.2}$$

where *a*, *b* parametrize the measurements. In principle, this parametrization should include POVMs, but often for simplicity one restricts state-behaviors to projective measurements (to practice this notion, see Exercise 3.1). States that are equivalent up to local unitaries produce the same state-behavior. Indeed, if *ρ*˜ = *UρU*† with *U* = *UA* ⊗ *UB*, then Tr*(ρx a* ⊗ *y b)* = Tr*(ρ*˜˜ *x a* ⊗ ˜ *y b)* where ˜ *x a* = *UAx aU*† *A* and ˜ *y b* = *UBy bU*† *B*.

To prove that *ρ* is a nonlocal resource, it is sufficient to find the families of measurements that generate a nonlocal behavior. Conversely, to prove that *ρ* is not a nonlocal resource, one should prove that all the choices of measurements lead to behaviors that can be simulated with LVs. Thus, the assessment of nonlocality for a quantum resource *ρ* involves *a priori* a heuristic step: Finding the suitable measurements or constructing an LV model. That being said, the nonlocality of two classes of states has been clarified with generic arguments:

• It follows immediately from the expression (3.1) that *separable states* can generate only local behaviors. Indeed, inserting *ρ* = - *k qkρk A* ⊗ *ρk B* into (3.1) one obtains *P(a*,*b*|*x*,*y)* = - *k qk*Tr*(ρk Ax a)*Tr*(ρk By b)*, which is of the form (2.16) for all measurements.

• We shall see in section 3.2.4 that *all pure entangled states* are nonlocal resources. The proof is constructive and provides a choice of measurements that generate a nonlocal behavior (without any claim of optimality, in any sense one may want to give to this word).

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As for the nonlocality of *mixed entangled states*, we shall review the approaches and main results in section 3.3.

Let us now turn to the properties of the *measurements*. If a player uses a set of jointlymeasurable1 POVMs, it will be possible to construct the joint probability distribution of all their outputs. In this case, that player can't contribute to a nonlocal behavior as proved in subsection 2.3.4. Conversely, in order to create a nonlocal behavior, it is a necessary to use non-jointly-measurable POVMs. One may ask whether this is also sufficient, but the answer is negative. Indeed, there are explicit examples of non-jointly-measurable POVMs for one player, such that the resulting behavior will be local, irrespective of the number and choice of measurements of the other player (Hirsch *et al*., 2018; Bene and Vértesi, 2018).

### **3.2 CHSH in Quantum Theory**

### **3.2.1 The CHSH Bell operator**

We shall use the notation *x*, *y* ∈ {0, 1} for the inputs and *a*, *b* ∈ {−1, +1} for the outputs, and work with the standard form (2.29) of the CHSH inequality. The correlation coefficients for given inputs (2.28) are now expressed as

$$\begin{split} E\_{\rm xy} &= \operatorname{Tr} \Big[ \rho \left( \boldsymbol{\Pi}\_{+1}^{\rm x} \otimes \boldsymbol{\Pi}\_{+1}^{\rm y} + \boldsymbol{\Pi}\_{-1}^{\rm x} \otimes \boldsymbol{\Pi}\_{-1}^{\rm y} - \boldsymbol{\Pi}\_{+1}^{\rm x} \otimes \boldsymbol{\Pi}\_{-1}^{\rm y} - \boldsymbol{\Pi}\_{-1}^{\rm x} \otimes \boldsymbol{\Pi}\_{+1}^{\rm y} \right) \Big] \\ &= \operatorname{Tr} \Big[ \rho \left( \boldsymbol{\Pi}\_{+1}^{\rm x} - \boldsymbol{\Pi}\_{-1}^{\rm x} \right) \otimes \left( \boldsymbol{\Pi}\_{+1}^{\rm y} - \boldsymbol{\Pi}\_{-1}^{\rm y} \right) \Big] \equiv \operatorname{Tr} \Big[ \rho A\_{\rm x} \otimes B\_{\rm y} \Big], \end{split} \tag{3.3}$$

where *Ax* = *x* +1 − *x* −1 and *By* = *y* +1 − *y* −1 are Hermitian operators of unspecified dimensions, whose eigenvalues lie between −1 and +1, and are equal to these values in the case of projective measurements. The value of the CHSH expression (2.29) in state *ρ* can be seen as the average value *S* = Tr(*ρ***S**) of the *CHSH operator*

$$\mathbf{S} = A\_0 \otimes B\_0 + A\_0 \otimes B\_1 + A\_1 \otimes B\_0 - A\_1 \otimes B\_1. \tag{3.4}$$

In particular, *a quantum state ρ violates CHSH if and only if there exist measurements such that* Tr(*ρ***S**) *>* 2.

Before continuing, we stress that a Bell test does *not* consist of single-shot measurements of a Bell operator. The Bell operator is self-adjoint, and as such it constitutes

1 The textbook equivalence between joint measurability and commutation holds only for projective measurements. For POVMs, commutation implies joint measurability but the converse is not true: There exist POVMs whose elements do not commute but are nonetheless jointly measurable (more in Appendix C).

a valid quantum observable. But it cannot be measured by separated players because (as can be expected and we shall prove in the next subsection) some of its eigenstates may be entangled. The measurements of the players are still described by the POVM elements *x a* and *y b*. The operators *Ax*, the *By* and **S** are derived from those and used for mathematical convenience.

### **3.2.2 The Tsirelson bound**

We proceed now to prove the famous *Tsirelson*2 *bound* (Cirel'son, 1980):

OUP CORRECTED PROOF – FINAL, 20/6/2019, SPi

$$\mathcal{S} \le 2\sqrt{2} \quad \text{for quantum behaviors.} \tag{3.5}$$

The bound is derived as an upper bound for the largest eigenvalue of **S** in absolute value, ||**S**||∞ = max*ψ* | *ψ***S***ψ*|. A very direct proof follows from noticing3 that **S** can be rewritten as

$$\mathbf{S} = \frac{1}{\sqrt{2}} (A\_0^2 + A\_1^2 + B\_0^2 + B\_1^2) - \frac{\sqrt{2} - 1}{8} [\text{positive operator}],\tag{3.6}$$

where the positive operator is given by

$$\begin{aligned} &\left((\sqrt{2}+1)(A\_0-B\_0)+A\_1-B\_1\right)^2+\left((\sqrt{2}+1)(A\_0-B\_1)-A\_1-B\_0\right)^2\\ &+\left((\sqrt{2}+1)(A\_1-B\_0)+A\_0+B\_1\right)^2+\left((\sqrt{2}+1)(A\_1+B\_1)-A\_0-B\_0\right)^2. \end{aligned}$$