The interpretation of the violation of Bell-Clauser-Horne inequalities is revisited, in relation with the notion of extension of QM predictions to unmeasurable correlations. Such extensions are compatible with QM predictions in many cases, in particular for observables with compatibility relations described by tree graphs. This implies classical representability of any set of correlations 〈Ai〉, 〈B〉, 〈AiB〉, and the equivalence of the Bell-Clauser-Horne inequalities to a non void intersection between the ranges of values for the unmeasurable correlation 〈A1A2 〉 associated to different choices for B. The same analysis applies to the Hardy model and to the “perfect correlations ” discussed by Greenberger, Horne, Shimony and Zeilinger. In all the cases, the dependence of an unmeasurable correlation on a set of variables allowing for a classical representation is the only basis for arguments about violations of locality and causality. 1 ar
The interpretation of the violation of Bell-Clauser-Horne inequalities is revisited, in relation with the notion of extension of QM predictions to unmeasurable correlations. Such extensions are compatible with QM predictions in many cases, in particular for observables with compatibility relations described by tree graphs. This implies classical representability of any set of correlations 〈Ai〉, 〈B〉, 〈AiB〉, and the equivalence of the Bell-Clauser-Horne inequalities to a non void intersection between the ranges of values for the unmeasurable correlation 〈A1A2 〉 associated to different choices for B. The same analysis applies to the Hardy model and to the “perfect correlations ” discussed by Greenberger, Horne, Shimony and Zeilinger. In all the cases, the dependence of an unmeasurable correlation on a set of variables allowing for a classical representation is the only basis for arguments about violations of locality and causality. 1 ar
The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
Violation of Bell inequalities is widely regarded as a definitive test for non-locality. However, Bell correlational inequalities must always be satisfied by all jointly present, cross-correlated data. The correlations of variable pairs obtained in repeated runs are not cross-correlated in this way and are not required to satisfy the Bell inequality. In addition, by using information regarded as non-local, proper joint correlations may be computed among counterfactual and measured variables. These correlations satisfy the Bell inequality, but are spatially non-stationary in angle. By using a simple symmetry condition, such considerations may be extended to inequalities in probabilities. The latter may be derived directly from correlational inequalities developed by Clauser, Horne, Shimony and Holt (CHSH). Violation of either correlational or probabilistic Bell inequalities then implies that the Bell correlation cannot be accounted for by a stochastic process that is spatially stationary in angle coordinates. However, other processes may still be allowed.
Quantum correlations that violate the Bell inequality cannot be explained by any local-hidden-variable theory that is measurement independent. However, this violation merely signifies the incompatibility of the underlying assumptions of reality, locality, and measurement independence, without providing a quantitative measure of the extent to which each assumption is violated. In contrast, Hall introduced a measure for each of the following assumptions: indeterminism, signaling, and measurement dependence, and Hall also generalized the Bell-Clauser-Horne-Shimony-Holt inequality, which provides a quantitative trade-off relationship between these assumptions. In this paper, we consider the introduction of hidden variables to be an essential assumption of Bell's theorem and introduce a quantification of hidden variables, which we term ``hiddenness.'' We derive a trade-off relation between hiddenness and measurement dependence that applies to any local-hidden-variable theory.
We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question of whether every proper (facet-defining) Bell inequality for two parties, other than the trivial ones from positivity, normalization, and no-signaling, can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games. We then show that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the nonlocal computation games, do not constitute facet-defining Bell inequalities, not even for the correlation polytope. We also extend this to an arbitrary prime number of outcomes for a specific class of these games. We then study the faces in the simplest Clauser-Horne-Shimony-Holt Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario. Finally, we relate the noncontextual polytope of single-party correlation inequalities with the cut polytope $\text{CUT}(\mathbf{\ensuremath{\nabla}}G)$, where G denotes the compatibility graph of observables in the contextuality scenario and $\mathbf{\ensuremath{\nabla}}G$ denotes the suspension graph of G. We observe that there exist facet-defining noncontextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the consistent exclusivity principle.
An inequality is deduced from Einstein's locality and a supplementary assumption. This inequality defines an experiment that can actually be performed with present technology to test local realism. Quantum mechanics violates this inequality a factor of 1.5. In contrast, quantum mechanics violates previous inequalities [for example, those of J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969); S. J. Freedman and I. F. Clauser ibid. 28, 938 (1972); and J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526 (1974)] by a factor of 2. Thus the magnitude of violation of the inequality derived in this paper is approximately 20.7% larger than the magnitude of violation of previous inequalities. This result can be particularly important for the experimental test of locality.