Table of contents

Many of the heated arguments about the meaning of 'Bellʼs theorem' arise because this phrase can refer to two different theorems that John Bell proved, the first in 1964 and the second in 1976. His 1964 theorem is the incompatibility of quantum phenomena with the dual assumptions of locality and determinism. His 1976 theorem is the incompatibility of quantum phenomena with the unitary property of local causality. This is contrary to Bellʼs own later assertions, that his 1964 theorem began with that single, and indivisible, assumption of local causality (even if not by that name). While there are other forms of Bellʼs theorems—which I present to explain the relation between Jarrett-completeness, 'fragile locality', and EPR-completeness—I maintain that Bellʼs two versions are the essential ones. Although the two Bellʼs theorems are logically equivalent, their assumptions are not, and the different versions of the theorem suggest quite different conclusions, which are embraced by different communities. For realists, the notion of local causality, ruled out by Bellʼs 1976 theorem, is motivated implicitly by Reichenbachʼs principle of common cause and explicitly by the principle of relativistic causality, and it is the latter which must be forgone. Operationalists pay no heed to Reichenbachʼs principle, but wish to keep the principle of relativistic causality, which, bolstered by an implicit 'principle of agent-causation', implies their notion of locality. Thus for operationalists, Bellʼs theorem is the 1964 one, and implies that it is determinism that must be forgone. I discuss why the two 'camps' are drawn to these different conclusions, and what can be done to increase mutual understanding.

While entanglement and violation of Bell inequalities were initially thought to be equivalent quantum phenomena, we now have different examples of entangled states whose correlations can be described by local hidden-variable models and, therefore, do not violate any of the Bell inequalities. We provide an up-to-date overview of the existing literature regarding local hidden-variable models for entangled quantum states, in both the bipartite and multipartite cases, and discuss some of the most relevant open questions in this context. Our review covers twenty five years of this line of research, beginning with the seminal work by Werner (1989 Phys. Rev. A 40 8), which provided the first example of an entangled state with a local model. Wernerʼs work, in turn, appeared twenty five years after the seminal work by Bell (1964 Physics1 195), about the impossibility of recovering the predictions of quantum mechanics using a local hidden-variable theory.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Bell inequalities are intended to show that local realist theories cannot describe the world. A local realist theory is one where physical properties are defined prior to and independent of measurement, and no physical influence can propagate faster than the speed of light. Quantum-mechanical predictions for certain experiments violate the Bell inequality while a local realist theory cannot, and this shows that a local realist theory cannot give those quantum-mechanical predictions. However, because of unexpected circumstances or 'loopholes' in available experiment tests, local realist theories can reproduce the data from these experiments. This paper reviews such loopholes, what effect they have on Bell inequality tests, and how to avoid them in experiment. Avoiding all these simultaneously in one experiment, usually called a 'loophole-free' or 'definitive' Bell test, remains an open task, but is very important for technological tasks such as device-independent security of quantum cryptography, and ultimately for our understanding of the world.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

This brief review discusses the problem of determining whether a given quantum state is separable or entangled. I describe an established approach to this problem that is based on the monogamy of entanglement, which is the observation that a pair of quantum systems that are strongly entangled must be uncorrelated with the rest of the world. Unentangled states on the other hand involve correlations that can be shared with many other parties. Checking whether a given quantum state is shareable involves constructing certain symmetric quantum state extensions and I discuss how to do this using a class of optimizations known as semidefinite programs. An attractive feature of this approach is that it generates explicit entanglement witnesses that can be measured to demonstrate the entanglement experimentally. In recent years analysis of this approach has greatly increased our understanding of the complexity of determining whether a given quantum state is entangled and this review aims to give a unified discussion of these developments. Specifically, I describe how to use finite quantum de Finetti theorems to prove that highly shareable states are nearly separable and use these results to understand the computational complexity of the problem.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

We present an overview of the quantitative theory of single-copy entanglement in finite-dimensional quantum systems. In particular we emphasize the point of view that different entanglement measures quantify different types of resources, which leads to a natural interdependence of entanglement classification and quantification. Apart from the theoretical basis, we outline various methods for obtaining quantitative results on arbitrary mixed states.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

We summarize important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger–Horne–Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramér–Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

I present my encounter with John Bell at CERN, our collaboration and joint work in particle physics. I also recall our quantum debates and give my personal view on Bellʼs fundamental work on quantum theory, in particular, on contextuality and nonlocality of quantum physics. Some mathematical and geometric aspects of entanglement are discussed as influence of Bellʼs theorem. Finally, I make some historical comments on the experimental side of Bell inequalities.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

It is shown that a main source of conflict between Einstein and the mainstream quantum physicists was his insistence that wave functions, like classical probability distributions, do not refer to individual particles and, in particular, do not describe individual systems completely. The EPR paper was written to argue for this position. By aiming at showing that wave functions are unsuitable as local hidden variables, the authors failed to see that a slight extension could have ruled out such local hidden variables in general. As background for this analysis of the EPR argument the notion of steering is described, and a version of the Bell argument is proved which emphasizes non-local signalling aspects. Finally, some background is given concerning a well-known paper by the present author, which is celebrating 25 years this year, and in which the first non-steering models were constructed.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Bellʼs theorem is 50 years old. There still remains a controversy about its implications. Much of this controversy has its roots in confusion regarding the premises from which the theorem can be derived. Some claim that a derivation of Bellʼs inequalities requires just a locality assumption and nothing more. Violations of the inequalities are then interpreted as a 'non-locality' or 'quantum non-locality'. We show that such claims are unfounded and that every derivation of Bellʼs inequalities requires a premise—in addition to locality and freedom of choice—which is either assumed tacitly, or unconsciously, or is embedded in a single compound condition (such as Bellʼs 'local causality'). The premise is equivalent to the assumption of the existence of additional variables which do not appear in the quantum formalism (in the form of determinism, joint probability for outcomes of all conceivable measurements, 'additional causes', 'hidden variables', 'complete description of the state' or counterfactual definiteness, etc). A certain irony is that perhaps the main message of the violation of Bellʼs inequalities is that our notion of locality should be based on an operationally well-defined no-signalling condition, rather than on local causality.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

On the 50th anniversary of Bell's monumental 1964 paper, there is still widespread misunderstanding about exactly what Bell proved. This misunderstanding derives in turn from a failure to appreciate the earlier argument of Einstein, Podolsky and Rosen. I retrace the history and logical structure of these arguments in order to clarify the proper conclusion, namely that any world that displays violations of Bell's inequality for experiments done far from one another must be non-local. Since the world we happen to live in displays such violations, actual physics is non-local.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

According to Maudlin, Bell showed that it is the World which is non-lcoal, and not just some particular theories of it. I argue that this conclusion is arrived at by taking for granted all assumptions of realism or 'classicality'. If these are taken into account the resulting conclusion that 'a classical world which allows for Bell inequality violations must be non-local' is in good agreement with the mainstream perception of Bellʼs theorem(s).

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

The insight due to John Bell that the joint behavior of individually measured entangled quantum systems cannot be explained by shared information remains a mystery to this day. We describe an experiment, and its analysis, displaying non-locality of entangled qutrit pairs. The non-locality of such systems, as compared to qubit pairs, is of particular interest since it potentially opens the door for tests of bipartite non-local behavior independent of probabilistic Bell inequalities, but of deterministic nature.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

The characterization of quantum correlations in terms of information-theoretic resource has been a fruitful approach to understand the power of quantum correlations as a resource. While bipartite entanglement and Bell inequality violation in this setting have been extensively studied, relatively little is known about their multipartite counterpart. In this paper, we apply and adapt the recently proposed definitions of multipartite nonlocality (Bancal et al 2013 Phys. Rev. A 88 014102) to the three- and four-partite scenarios to gain new insight on the resource aspect of multipartite nonlocal quantum correlations. Specifically, we show that reproducing certain tripartite quantum correlations requires mixtures of classical resources—be it the ability to change the groupings or the time orderings of measurements. Thus, when seen from the perspective of biseparable one-way classical signaling resources, certain tripartite quantum correlations do not admit a definite causal order. In the four-partite scenario, we obtain a superset description of the set of biseparable correlations which can be produced by allowing two groups of bipartite non-signaling resources. Quantum violation of the resulting Bell-like inequalities is investigated. As a byproduct, we obtain some new examples of device-independent witnesses for genuine four-partite entanglement, and also device-independent witnesses that allow one to infer the structure of the underlying multipartite entanglement.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

We consider a subclass of bipartite CHSH-type Bell inequalities. We investigate operations which leave their Tsirelson bound invariant, but change their classical bound. The optimal observables are unaffected except for a relative rotation of the two laboratories. We illustrate the utility of these operations by giving explicit examples. We prove that, for a fixed quantum state and fixed measurement setup except for a relative rotation of the two laboratories, there is a Bell inequality that is maximally violated for this rotation, and we optimize some Bell inequalities with respect to the maximal violation. Finally, we optimize the qutrit to qubit ratio of some dimension witnessing Bell inequalities.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

It is known that the local bound of a Bell inequality is sensitive to the knowledge of the external observer about the settings statistics. Here we ask how that sensitivity depends on the structure of that knowledge. It turns out that in some cases it may happen that the local bound is much more sensitive to the adversaryʼs knowledge about the settings of one party than the other. Remarkably, there are Bell inequalities which are highly asymmetric with respect to the adversaryʼs knowledge about local settings. This property may be viewed as a hidden intrinsic asymmetry of Bell inequalities. Potential implications of the revealed asymmetry effect are also discussed.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

With the advent of device-independent quantum information processing, nonlocality is nowadays regarded as a resource that can be used for various tasks. Using the analogy of entanglement theory, we approach nonlocality from this perspective. In order to do so, we analyze in full detail the operations that can be implemented in this scenario and under which nonlocality cannot increase. This provides a theoretical basis from which to study how to order and quantify nonlocal behavior. Finally, we review several nonlocality measures and discuss their validity from this point of view.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Bellʼs 1964 theorem causes a severe problem for the notion that correlations require explanation, encapsulated in Reichenbachʼs principle of common cause. Despite being a hallmark of scientific thought, dropping the principle has been widely regarded as much less bitter medicine than the perceived alternative—dropping relativistic causality. Recently, however, some authors have proposed that modified forms of Reichenbachʼs principle could be maintained even with relativistic causality. Here we break down Reichenbachʼs principle into two independent assumptions—the principle of common cause proper and factorization of probabilities. We show how Bellʼs theorem can be derived from these two assumptions plus relativistic causality and the law of total probability for actual events, and we review proposals to drop each of these assumptions in light of the theorem. In particular, we show that the non-commutative common causes of Hofer-Szabó and Vecsernyés fail to have an analogue of the notion that the common causes can explain the observed correlations. Moreover, we show that their definition can be satisfied trivially by any quantum product state for any quantum correlations. We also discuss how the conditional states approach of Leifer and Spekkens fares in this regard.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Bell's theorem started with two qubits (spins 1/2). It is a 'no-go' statement on classical (local causal) models of quantum correlations. After 25 years, it turned out that for three qubits the situation is even more astonishing. General statements concerning higher dimensional systems, qutrits, etc, started to appear even later, once the picture with spin (higher than 1/2) was replaced by a broader one, allowing all possible observables. This work is a continuation of the Gdansk effort to take advantage of the fact that Bell's theorem can be put in the form of a linear programming problem, which in turn can be translated into a computer code. Our results are numerical and classify the strength of the violation of local causality by various families of three-qutrit states, as measured by the resistance to noise. This is previously uncharted territory. The results may be helpful in suggesting which three-qutrit states will be handy for applications in quantum information protocols. One of the surprises is that the W state turns out to reveal a stronger violation of local causality than the GHZ (Greenberger–Horne–Zeilinger) state.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

The correlations that admit a local hidden-variable model are described by a family of polytopes, whose facets are the Bell inequalities. The Clauser–Horne–Shimony–Holt (CHSH) inequality is the simplest such Bell inequality and is a facet of every Bell polytope. We investigate for which Bell polytopes the CHSH inequality is also the unique (non-trivial) facet. We prove that the CHSH inequality is the unique facet for all bipartite polytopes where at least one party has a binary choice of dichotomic measurements, irrespective of the number of measurement settings and outcomes for the other party. Based on numerical results, we conjecture that it is also the unique facet for all bipartite polytopes involving two measurements per party where at least one measurement is dichotomic. Finally, we remark that these two situations can be the only ones for which the CHSH inequality is the unique facet, i.e., any polytope that does not correspond to one of these two cases necessarily has facets that are not of the CHSH form. As a byproduct of our approach, we derive a new family of facet inequalities.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Non-contextuality (NC) and Bell inequalities can be expressed as bounds Ω for positive linear combinations S of probabilities of events, $S\leqslant \Omega$. Exclusive events in S can be represented as adjacent vertices of a graph called the exclusivity graph of S. In the case that events correspond to the outcomes of quantum projective measurements, quantum probabilities are intimately related to the Grötschel–Lovász–Schrijver theta body of the exclusivity graph. Then, one can easily compute an upper bound to the maximum quantum violation of any NC or Bell inequality by optimizing S over the theta body and calculating the Lovász number of the corresponding exclusivity graph. In some cases, this upper bound is tight and gives the exact maximum quantum violation. However, in general, this is not the case. The reason is that the exclusivity graph does not distinguish among the different ways exclusivity can occur in Bell-inequality (and similar) scenarios. An interesting question is whether there is a graph-theoretical concept which accounts for this problem. Here we show that, for any given N-partite Bell inequality, an edge-coloured multigraph composed of N single-colour graphs can be used to encode the relationships of exclusivity between each partyʼs parts of the events. Then, the maximum quantum violation of the Bell inequality is exactly given by a refinement of the Lovász number that applies to these edge-coloured multigraphs. We show how to calculate upper bounds for this number using a hierarchy of semi-definite programs and calculate upper bounds for I₃, I₃₃₂₂ and the three bipartite Bell inequalities whose exclusivity graph is a pentagon. The multigraph-theoretical approach introduced here may remove some obstacles in the program of explaining quantum correlations from first principles.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Since John S Bell demonstrated the interest of studying linear combinations of probabilities in relation with the EPR paradox in 1964, Bell inequalities have lead to numerous developments. Unfortunately, the description of Bell inequalities is subject to several degeneracies, which make any exchange of information about them unnecessarily hard. Here, we analyze these degeneracies and propose a decomposition for Bell-like inequalities based on a set of reference expressions which is not affected by them. These reference expressions set a common ground for comparing Bell inequalities. We provide algorithms based on finite group theory to compute this decomposition. Implementing these algorithms allows us to set up a compendium of reference Bell-like inequalities, available online at www.faacets.com. This website constitutes a platform where registered Bell-like inequalities can be explored, new inequalities can be compared to previously-known ones and relevant information on Bell inequalities can be added in a collaborative manner.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Bell showed 50 years ago that quantum theory is non-local via his celebrated inequalities, turning the issue of quantum non-locality from a matter of taste into a matter of test. Years later, Hardy proposed a test for non-locality without inequality, which is a kind of 'something-versus-nothing' argument. Hardyʼs test for n particles induces an n-partite Bellʼs inequality with two dichotomic local measurements for each observer, which has been shown to be violated by all entangled pure states. Our first result is to show that the Bell–Hardy inequality arising form Hardyʼs non-locality test is tight for an arbitrary number of parties, i.e., it defines a facet of the Bell polytope in the given scenario. On the other hand quantum theory is not that non-local since it forbids signaling and even not as non-local as allowed by non-signaling conditions, i.e., quantum mechanical predictions form a strict subset of the so called non-signaling polytope. In the scenario of each observer measuring two dichotomic observables, Fritz established a duality between the Bell polytope and the non-signaling polytope: tight Bellʼs inequalities, the facets of the Bell polytope, are in a one-to-one correspondence with extremal non-signaling boxes, the vertices of the non-signaling polytope. Our second result is to provide an alternative and more direct formula for this duality. As an example, the tight Bell–Hardy inequality gives rise to an extremal non-signaling box that serves as a natural multipartite generalization of Popescu–Rohrlich box.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems. The known constructions of multipartite Bell inequalities contain, however, correlation functions involving all observers, making their experimental implementation difficult. The main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest-to-measure quantities, that is, the two-body correlations. In particular, we determine all three- and four-partite Bell inequalities constructed from one- and two-body expectation values that obey translational symmetry, and show that they reveal nonlocality in multipartite states. Also, by providing a particular example of a five-partite Bell inequality, we show that nonlocality can be detected from two-body correlators involving only nearest neighbours. Finally, we demonstrate that any translationally invariant Bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites. We provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant Bell inequality.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

The 'Werner gap' is the range of relevant parameters characterizing a quantum state for which it is both entangled and admits a local hidden variable model. Werner showed that the gap becomes maximal for entanglement mixed with white noise if subsystems have infinitely many levels. Here we study pure entangled states mixed with simple coloured noise modelled as a single pure product state. We provide an explicit local hidden variable model for quantum correlations of some states of this family and provide hints that there is probably a model for all quantum predictions. This demonstrates essentially a maximal Werner gap already for just two qubits. Additionally to its fundamental interest, the study has implications for quantum computation and communication.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Bell's theorem is a fundamental theorem in physics concerning the incompatibility between some correlations predicted by quantum theory and a large class of physical theories. In this paper, we introduce the hypothesis of accountability, which demands that it is possible to explain the correlations of the data collected in many runs of a Bell experiment in terms of what happens in each single run. Under this assumption, and making use of a recent result by Colbeck and Renner (2011 Nature Commun.2 411), we then show that any nontrivial account of these correlations in the form of an extension of quantum theory must violate parameter independence. Moreover, we analyze the violation of outcome independence of quantum mechanics and show that it is also a manifestation of nonlocality.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Bellʼs inequality fundamentally changed our understanding of quantum mechanics. Bellʼs insight that non-local correlations between quantum systems cannot be explained classically can be verified experimentally, and has numerous applications in modern quantum information. Today, the Clauser–Horne–Shimony–Holt (CHSH) inequality is probably the most well-known Bell inequality and it has given us a wealth of understanding in what differentiates the classical from the quantum world. Yet, there are certainly other means of quantifying 'Bell non-locality without inequalities' such as the famous Hardyʼs paradox. As such, one may wonder whether these are entirely different approaches to non-locality. For this anniversary issue, we unify the perspective of the CHSH inequality and Hardy's paradox into one family of non-local games which include both as special cases.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

The amount of intrinsic randomness that can be extracted from measurement on quantum systems depends on several factors: notably, the power given to the adversary and the level of characterization of the devices of the authorized partners. After presenting a systematic introduction to these notions, in this paper we work in the class of least adversarial power, which is relevant for assessing setups operated by trusted experimentalists, and compare three levels of characterization of the devices. Many recent studies have focused on the so-called 'device-independent' level, in which a lower bound on the amount of intrinsic randomness can be certified without any characterization. The other extreme is the case when all the devices are fully characterized: this 'tomographic' level has been known for a long time. We present for this case a systematic and efficient approach to quantifying the amount of intrinsic randomness, and show that setups involving ancillas (positive-operator valued measures, pointer measurements) may not be interesting here, insofar as one may extract randomness from the ancilla rather than from the system under study. Finally, we study how much randomness can be obtained in presence of an intermediate level of characterization related to the task of 'steering', in which Bobʼs device is fully characterized while Aliceʼs is a black box. We obtain our results here by adapting the NPA hierarchy of semidefinite programs to the steering scenario.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

We investigate the conditions under which a set of multipartite nonlocal correlations can describe the distributions achievable by distant parties conducting experiments in a consistent universe. Several questions are posed, such as: are all such sets 'nested', i.e., contained into one another? Are they discrete or do they form a continuum? How many of them are supraquantum? Are there non-trivial polytopes among them? We answer some of these questions or relate them to established conjectures in complexity theory by introducing a 'zoo' of physically consistent sets, which can be characterized efficiently via either linear or semidefinite programming. As a bonus, we use the zoo to derive, for the first time, concrete impossibility results in nonlocality distillation.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Acin et al (2010 Phys. Rev. Lett.104 140404) introduced a unified framework for the study of no-signalling correlations. Such a framework is based on the notion of local quantum measurements, but, in order to account for beyond-quantum correlations, global pseudo-states that are not positive semidefinite are allowed. After a short review of the formalism, we consider its use in the quantification of both general non-local and beyond-quantum correlations. We argue that the unified framework for correlations provides a simple approach to such a quantification, in particular when the quantification is meant to be operational and meaningful in a resource-theory scenario, i.e., when considering the processing of resources by means of non-resources. We relate different notions of robustness of correlations, both at the level of (pseudo-)states and abstract probability distributions, with particular focus on the beyond-quantum robustness of correlations and pseudo-states. We revisit known results and argue that, within the unified framework, the relation between the two levels—that of operators and that of probability distributions—is very strict. We point out how the consideration of robustness at the two levels leads to a natural framework for the quantification of entanglement in a device-independent way. Finally, we show that the beyond-quantum robustness of the non-positive operators needed to achieve beyond-quantum correlations coincides with their negativity and their distance from the set of quantum states. As an example, we calculate the beyond-quantum robustness for the case of a noisy Popescu–Rohrlich box.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

The possibility of experimentally testing the Bell–Kochen–Specker theorem is investigated critically, following the demonstrations by Meyer, Kent, and Clifton–Kent that the predictions of quantum mechanics are indistinguishable (up to arbitrary precision) from those of a non-contextual model, and the subsequent debate about the extent to which these models are actually classical or non-contextual. The present analysis starts from a careful consideration of these 'finite-precision' approximations. A stronger condition for non-contextual models, dubbed ontological faithfulness, is exhibited. It is shown that this allows us to approximately formulate the constraints in Bell–Kochen–Specker theorems, such as to render the usual proofs robust. Consequently, one can experimentally test to finite precision ontologically faithful non-contextuality, and thus experimentally refute explanations from this smaller class. We include a discussion of the relation of ontological faithfulness to other proposals to overcome the finite precision objection.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

The Franson interferometer, proposed in 1989 (Franson 1989 Phys. Rev. Lett.62 2205–08), beautifully shows the counter-intuitive nature of light. The quantum description predicts sinusoidal interference for specific outcomes of the experiment, and these predictions can be verified in experiment. In the spirit of Einstein, Podolsky, and Rosen it is possible to ask if the quantum-mechanical description (of this setup) can be considered complete. This question will be answered in detail in this paper, by delineating the quite complicated relation between energy-time entanglement experiments and Einstein–Podolsky–Rosen (EPR) elements of reality. The mentioned sinusoidal interference pattern is the same as that giving a violation in the usual Bell experiment. Even so, depending on the precise requirements made on the local realist model, this can imply (a) no violation, (b) smaller violation than usual, or (c) full violation of the appropriate statistical bound. Alternatives include (a) using only the measurement outcomes as EPR elements of reality, (b) using the emission time as EPR element of reality, (c) using path realism, or (d) using a modified setup. This paper discusses the nature of these alternatives and how to choose between them. The subtleties of this discussion needs to be taken into account when designing and setting up experiments intended to test local realism. Furthermore, these considerations are also important for quantum communication, for example in Bell-inequality-based quantum cryptography, especially when aiming for device independence.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

We provide a class of Bell diagonal entanglement witnesses displaying an additional local symmetry—a maximal commutative subgroup of the unitary group U(n). Remarkably, this class of witnesses is parameterized by a torus being a maximal commutative subgroup of an orthogonal group $SO(n-1)$. It is shown that a generic element from the class defines an indecomposable entanglement witness. The paper provides a geometric perspective for some aspects of the entanglement theory and an interesting interplay between group theory and block-positive operators in ${{\mathbb{C}}^{n}}\otimes {{\mathbb{C}}^{n}}$.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Unextendible product bases (UPBs) have been shown to have many important uses in quantum information theory, particularly in the qubit case. However, very little is known about their mathematical structure beyond three qubits. We present several new results about qubit UPBs, including a complete characterization of all four-qubit UPBs, which we show there are exactly 1446 of. We also show that there exist p-qubit UPBs of almost all sizes less than 2^p.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

We derive a simple lower bound on the geometric measure of entanglement for mixed quantum states in the case of a general multipartite system. The main ingredient of the presented derivation is the triangle inequality applied to the root infidelity distance in the space of density matrices. The obtained bound leads to entanglement criteria with a straightforward interpretation. The proposed criteria provide an experimentally accessible, powerful entanglement test.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Non-local properties (globalness) of a non-separable unitary determine how the unitary affects the entanglement properties of a quantum state. We apply a given two-qubit unitary in a quadpartite system including two reference systems and analyze its local operations and classical communication (LOCC) partial invertibility under two-party LOCC. A decomposition given by Kraus and Cirac for two-qubit unitaries shows that globalness is completely characterized by three parameters. Our analysis shows that the number of non-zero parameters (the Kraus–Cirac number) has operational significance when converting entanglement properties of multipartite states. All two-qubit unitaries have a Kraus–Cirac number of at most 3, while those with at most 1 or 2 are equivalent, up to local unitaries, to a controlled unitary or matchgate, respectively. The presented operational framework distinguishes the unitaries with a Kraus–Cirac number of 2 and 3, which was not possible by the known measure of the operator Schmidt decomposition. We also analyze how the Kraus–Cirac number changes when two or more two-qubit unitaries are applied sequentially.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

The three-tangle is a measure of three-way entanglement in a system of three qubits. For a pure state, it can be understood as the residual entanglement not accounted for by pairwise entanglements between individual qubits. Here we define and evaluate the analogous quantity for three rebits (that is, binary systems in the real-amplitude variant of quantum theory). We find that the resulting formula is the same as in the complex case, except that an overall absolute value sign is missing. As a result, the rebit three-tangle can be negative, expressing the possibility of non-monogamous entanglement in real-amplitude quantum theory (for entanglement based on the convex-roof construction). We then relate the entanglement among three rebits to the entanglement of two qubits, by re-expressing the two-qubit state as a three-rebit state in the ubit model.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.