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arXiv:1612.06064 · Foundations of Physics

Bell’s Nonlocality in a General Nonsignaling Case: Quantitatively and Conceptually

Elena R. Loubenets

Published Mar 7, 2017• 17 citations
arXivHTMLPDF
DOI
10.1007/s10701-017-0077-4

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Bell's theorem shows that local measurements on entangled states give rise to correlations incompatible with local-hidden-variable models. The degree of quantum nonlocality is not maximal though, as there are even more nonlocal theories beyond quantum theory still compatible with the nonsignaling principle. In spite of decades of research, we still have a very fragmented picture of the whole geometry of these different sets of correlations. Here we employ both analytical and numerical tools to ameliorate that. First, we identify two different classes of Bell scenarios where the nonsignaling correlations can behave very differently: In one case, the correlations are generically quantum and nonlocal while in the other quite the opposite happens as the correlations are generically classical and local. Second, by randomly sampling over nonsignaling correlations, we compute the distribution of a nonlocality quantifier based on the trace distance to the local set. With that we conclude that the nonlocal correlations can show a concentration phenomenon: Their distribution is peaked at a distance from the local set that increases both with the number of parts or measurements being performed.
Inspired by the recent works of Foster et al. [Phys. Rev. Lett. 102, 120401 (2009)] and Brunner et al. [Phys. Rev. Lett. 102, 160403 (2009)], we present a nonlocality distillation protocol for two three-level (qutrit) systems in the framework of generalized nonsignaling theories. Our protocol is based on a three-setting Bell inequality. It works efficiently for a specific class of three-input--three-output nonlocal boxes. In the asymptotic limit, all these nonlocal boxes can be distilled to the maximally nonlocal box defined by the inequality and nonsignaling constraints. Then we introduce a contracting protocol that reduces these boxes to the so-called ``correlated nonlocal boxes.'' As a result, our three-input--three-output nonlocal boxes also make communication complexity trivial and appear very unlikely to exist in nature.