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arXiv:2004.12968

Bipartite non-locality beyond Bell inequalities by means of multivariable correlations

Bruno Leggio, Bruno Bellomo, Romain Azaïs, Przemysław Prusinkiewicz, Christophe Godin

Published Apr 27, 2020• 0 citations
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Abstract

We give a set of necessary conditions for locality in bipartite systems, which include and generalize known Bell's inequalities. Each condition corresponds to a specific order of the expansion of random variables defined on graphs, in terms of genuinely n-variable correlation functions. The first non-trivial order leads to known Bell inequalities, while higher orders produce additional, non-equivalent conditions. In particular, in CHSH settings, we obtain at least two additional tight conditions which do not reduce to the CHSH inequality. This shows that tight Bell inequalities are sufficient but not necessary conditions for the non-locality of bipartite quantum correlations.

DOI
10.48550/arxiv.2004.12968

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We study the nonlocality of arbitrary dimensional bipartite quantum states. By computing the maximal violation of a set of multi-setting Bell inequalities, an analytical and computable lower bound has been derived for general two-qubit states. This bound gives the necessary condition that a two-qubit state admits no local hidden variable models. The lower bound is shown to be better than that from the CHSH inequality in judging the nonlocality of some quantum states. The results are generalized to the case of high dimensional quantum states, and a sufficient condition for detecting the non-locality has been presented.
We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multipartite ddd-dimensional systems. All the Bell inequalities based on this generalization take the same simple form as the CHSH inequality. For small systems, numerical results show that the new inequalities are tight and we believe this is also valid for higher-dimensional systems. Moreover, the new inequalities are relevant to the previous ones and, for bipartite systems, our inequality is equivalent to the Collins-Gisin-Linden-Masser-Popescu inequality.
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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.