Abstract
Bell's theorem shows that no local hidden-variable model can explain the measurement statistics of a quantum system shared between two parties, thus ruling out a classical (local) understanding of nature. In this paper we demonstrate that by relaxing the positivity restriction in the hidden-variable probability distribution it is possible to derive quasiprobabilistic Bell inequalities whose sharp upper bound is written in terms of a negativity witness of said distribution. This provides an analytic solution for the amount of negativity necessary to violate the Clauser-Horne-Shimony-Holt inequality by an arbitrary amount, therefore revealing the amount of negativity required to emulate the quantum statistics in a Bell test.
Highlights
It has been 60 years since John Stewart Bell wrote his famous paper on the Einstein-Podolsky-Rosen (EPR) paradox [1], and 50 years since the first experimental Bell test [2]
This shows that it is possible to recapture the nonlocal features of Bell experiments through having a finite amount of negativity allowed in a hidden-variable distribution over scenarios which are, in themselves, entirely local and classical
We have shown that there exists a relationship between the amount of negativity allowed in a joint hidden-variable distribution, and the degree to which said distribution can demonstrate nonlocality in a Bell experiment
Summary
It has been 60 years since John Stewart Bell wrote his famous paper on the Einstein-Podolsky-Rosen (EPR) paradox [1], and 50 years since the first experimental Bell test [2]. Our main result is that the violation of the CHSH inequality (and n-measurement generalisations) can be exactly characterised by a negativity witness of the hidden-variable distribution defined over the local states, and that there exists quasiprobability distributions which can saturate (up to the no-signalling limit) any such violation, whilst still having well-defined local statistics. This shows that it is possible to recapture the nonlocal features of Bell experiments through having a finite amount of negativity allowed in a hidden-variable distribution over scenarios which are, in themselves, entirely local and classical. We are interested in the achievable bounds of a classical system’s probability distribution when the hidden-variable distribution in said probability distribution can be negative
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