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arXiv:1206.3212 · Physical Review A

All noncontextuality inequalities for the n -cycle scenario

Mateus Araújo, Marco Túlio Quintino, Costantino Budroni, Marcelo Terra Cunha, Adán Cabello

Published Aug 21, 2013• 156 citations
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Abstract

The problem of separating classical from quantum correlations is, in general, intractable and has been solved explicitly only in a few cases. In particular, known methods cannot provide general solutions for an arbitrary number of settings. We provide the complete characterization of the classical correlations and the corresponding maximal quantum violations for the case of $n\ensuremath{\ge}4$ observables ${X}_{0},...,{X}_{n\ensuremath{-}1}$, where each consecutive pair ${{X}_{i},{X}_{i+1}}$, sum $\mathrm{mod}\phantom{\rule{0.16em}{0ex}}n$, is jointly measurable. This generalizes both the Clauser-Horne-Shimony-Holt and the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky scenarios, which are the simplest ones for locality and noncontextuality, respectively. In addition, we provide explicit quantum states and settings with maximal quantum violation and minimal quantum dimension.

DOI
10.1103/physreva.88.022118

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The problem of separating classical from quantum correlations is, in general, intractable and has been solved explicitly only in a few cases. In particular, known methods cannot provide general solutions for an arbitrary number of settings. We provide the complete characterization of the classical correlations and the corresponding maximal quantum violations for the case of n\ensuremath≥4n\ensuremath{\ge}4n\ensuremath≥4 observables X0,...,Xn\ensuremath−1{X}_{0},...,{X}_{n\ensuremath{-}1}X0​,...,Xn\ensuremath−1​, where each consecutive pair Xi,Xi+1{{X}_{i},{X}_{i+1}}Xi​,Xi+1​, sum modn\mathrm{mod}\phantom{\rule{0.16em}{0ex}}nmodn, is jointly measurable. This generalizes both the Clauser-Horne-Shimony-Holt and the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky scenarios, which are the simplest ones for locality and noncontextuality, respectively. In addition, we provide explicit quantum states and settings with maximal quantum violation and minimal quantum dimension.
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 GGG denotes the compatibility graph of observables in the contextuality scenario and $\mathbf{\ensuremath{\nabla}}G$ denotes the suspension graph of GGG. 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.
The assumption of local realism, in a Bell locality scenario, imposes nontrivial conditions on the Shannon entropies of the associated probability distributions, expressed by linear entropic Bell inequalities. In principle, these entropic inequalities provide necessary but not sufficient criteria for the existence of a local hidden variable model reproducing the correlations, as, for example, the paradigmatic nonlocal Popescu-Rohrlich (PR) box is entropically not different from a classically correlated box. In this paper we show that for the nnn-cycle scenario, entropic inequalities completely characterize the set of local correlations. In particular, every nonsignaling box which violates the Clauser-Horne-Shimony-Holt (CHSH) inequality---including the PR box---can be locally modified so that it also violates the entropic version of CHSH inequality. As we show, any nonlocal probabilistic model when appropriately mixed with a local model, violates an entropic inequality, thus evidencing a very peculiar kind of nonlocality. As the nnn-cycle captures equally well both the notion of local realism introduced by Bell and that of noncontextuality presented by the Kochen-Specker theorem, the results are also valid for noncontextuality scenarios.
The information-theoretic approach to Bell's theorem is developed with use of the conditional qqq-entropies. The qqq-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional qqq-entropies for all q≥1q\geq1q≥1. Quantum violations of the new inequalities are exemplified within the Clauser--Horne--Shimony--Holt (CHSH) and Klyachko--Can--Binicio\v{g}lu--Shumovsky (KCBS) scenarios. An extension to the case of nnn-cycle scenario is briefly mentioned. The new inequalities with conditional qqq-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The qqq-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the qqq-entropic formulation.
In this paper we analyze the notion of measurement noncontextuality (MNC) and identify contextual scenarios which involve sequential measurements of only a single measurement device. We show that any noncontextual ontological model fails to explain the statistics of outcomes of a single carefully constructed positive-operator-valued measure (POVM) executed sequentially on a quantum system. The context of measurement arises from the different configurations in which the device can be used. We develop an inequality from the noncontextual (NC) ontic model, and construct a quantum situation involving measurements from the Klyachko-Can-Binicio\ifmmode \check{g}\else \v{g}\fi{}lu-Shumovsky (KCBS) inequality. We show that the resultant statistics arising from this device violate our NC inequality. This device can be generalized by incorporating measurements from arbitrary nnn-cycle contextuality inequalities of which n=5n=5n=5 corresponds to the KCBS inequality. We show that the NC and quantum bounds for various scenarios can be derived more easily using only the functional relationships between the outcomes for larger values nnn. This makes it one of the simpler contextual inequalities to analyze.
Abstract The Clauser–Horne–Shimony-Holt (CHSH) inequality tests locality in quantum theory and is recognized as one of Bell’s inequalities. In contrast, the Klyachko–Can–Binicioğlu–Shumovsky (KCBS) inequality tests noncontextuality, which refers to the idea that measurement outcomes are determined by intrinsic properties independent of other measurements that could be performed simultaneously. While individual quantum states can violate these inequalities separately, it was believed that no single state could violate both simultaneously. This relationship suggests a trade-off between nonlocality and contextuality: if one is violated, the other cannot be. Such a relationship is known as a monogamy relation. Xue et al demonstrated, however, that simultaneous violations of both the CHSH and KCBS inequalities are possible with specific choices of observables. This finding challenges the universal validity of the monogamy relation. They further showed that numerous scenarios exist in which both the CHSH inequality and a noncontextuality inequality involving more than five observables are violated. Nevertheless, the number of such scenarios is finite. In this paper, we extend these findings in two significant ways. First, we show that infinitely many scenarios exist where both the CHSH inequality and a noncontextuality inequality involving an odd number of observables are violated. This demonstrates that the monogamy relation between the CHSH inequality and the noncontextuality inequality fails in infinitely many cases. Second, we identify quantum states that violate both the KCBS inequality and a locality inequality distinct from the CHSH inequality. Our results highlight that the so-called ‘monogamy relation’ between nonlocality and contextuality is not universally valid. Instead, it appears that both phenomena can coexist under broader conditions than initially expected, thus challenging the conventional viewpoint and paving the way for new theoretical and experimental research.
Tests for quantum contextuality in terms of qqq-entropies
The information-theoretic approach to Bell's theorem is developed with use of the conditional qqq-entropies. The qqq-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional qqq-entropies for all q≥1q\geq1q≥1. Quantum violations of the new inequalities are exemplified within the Clauser-Horne-Shimony-Holt (CHSH) and Klyachko-Can-Binicioǧlu-Shumovsky (KCBS) scenarios. An extension to the case of nnn-cycle scenario is briefly mentioned. The new inequalities with conditional qqq-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The qqq-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the qqq-entropic formulation.