Local hidden variable modelling, classicality, quantum separability, and the original Bell inequality

Abstract

We introduce a general condition sufficient for the validity of the original Bell inequality (1964) in a local hidden variable (LHV) frame. This condition can be checked experimentally and incorporates only as a particular case the assumption on perfect correlations or anticorrelations usually argued for this inequality in the literature. Specifying this general condition for a quantum bipartite case, we introduce the whole class of bipartite quantum states, separable and nonseparable, that (i) admit an LHV description under any bipartite measurements with two settings per site; (ii) do not necessarily exhibit perfect correlations and may even have a negative correlation function if the same quantum observable is measured at both sites but (iii) satisfy the "perfect correlation" version of the original Bell inequality for any three bounded quantum observables A, A'=B, B' at sites "A" and "B", respectively. Analysing the validity of this general LHV condition under classical and quantum correlation experiments with the same physical context, we stress that, unlike the Clauser-Horne-Shimony-Holt (CHSH) inequality, the original Bell inequality distinguishes between classicality and quantum separability.

Edited version

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• Local quasi hidden variable modelling and violations of Bell-type inequalities by a multipartite quantum state
• Evaluating the Maximal Violation of the Original Bell Inequality by Two-Qudit States Exhibiting Perfect Correlations/Anticorrelations
• Quantum analog of the original Bell inequality for two-qudit states with perfect correlations/anticorrelations
• Specifying nonlocality of a pure bipartite state and analytical relations between measures for bipartite nonlocality and entanglement