On monogamy of non-locality and macroscopic averages: examples and preliminary results

Abstract

We explore a connection between monogamy of non-locality and a weak macroscopic locality condition: the locality of the average behaviour. These are revealed by our analysis as being two sides of the same coin. Moreover, we exhibit a structural reason for both in the case of Bell-type multipartite scenarios, shedding light on but also generalising the results in the literature [Ramanathan et al., Phys. Rev. Lett. 107, 060405 (2001); Pawlowski & Brukner, Phys. Rev. Lett. 102, 030403 (2009)]. More specifically, we show that, provided the number of particles in each site is large enough compared to the number of allowed measurement settings, and whatever the microscopic state of the system, the macroscopic average behaviour is local realistic, or equivalently, general multipartite monogamy relations hold. This result relies on a classical mathematical theorem by Vorob'ev [Theory Probab. Appl. 7(2), 147-163 (1962)] about extending compatible families of probability distributions defined on the faces of a simplicial complex -- in the language of the sheaf-theoretic framework of Abramsky & Brandenburger [New J. Phys. 13, 113036 (2011)], such families correspond to no-signalling empirical models, and the existence of an extension corresponds to locality or non-contextuality. Since Vorob'ev's theorem depends solely on the structure of the simplicial complex, which encodes the compatibility of the measurements, and not on the specific probability distributions (i.e. the empirical models), our result about monogamy relations and locality of macroscopic averages holds not just for quantum theory, but for any empirical model satisfying the no-signalling condition. In this extended abstract, we illustrate our approach by working out a couple of examples, which convey the intuition behind our analysis while keeping the discussion at an elementary level.

In Proceedings QPL 2014, arXiv:1412.8102

References in corpus (8)

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• The Sheaf-Theoretic Structure Of Non-Locality and Contextuality
• Logical Bell Inequalities
• Monogamy of Bell's inequality violations in non-signaling theories
• The Cohomology of Non-Locality and Contextuality
• Testing a Bell inequality in multi-pair scenarios
• Extendability in the Sheaf-theoretic Approach: Construction of Bell Models from Kochen-Specker Models
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• Kochen-Specker Contextuality
• Necessary and sufficient condition for contextuality from incompatibility
• On the Cohomology of Contextuality
• Macroscopically nonlocal quantum correlations
• Closing Bell: Boxing black box simulations in the resource theory of contextuality
• Continuous-variable nonlocality and contextuality
• Combining contextuality and causality: a game semantics approach