All separable states satisfy all Bell-type inequalities, which involve as their assumption only existence of local realistic (local hidden variable) models of the correlations of spatially separated systems, observed by two or more observers making independent decisions on what to measure (free will). The recent observation by Loubenets, that some separable states do not satisfy the original Bell inequality (1964) has no consequences whatsoever for the studies of the relation of separability with local realism. The original Bell inequality was derived using an additional assumption that the local results for a certain pair of local settings reveal perfect Einstein–Podolsky–Rosen (EPR) correlations. Therefore violation of this inequality by some quantum predictions implies that either (i) the predictions do not allow a local realistic model, or (ii) the predictions do not have the required EPR correlations, or finally both (i) and (ii).
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Żukowski, M. Separability of Quantum States vs. Original Bell (1964) Inequalities. Found Phys 36, 541–545 (2006). https://doi.org/10.1007/s10701-005-9038-4
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DOI: https://doi.org/10.1007/s10701-005-9038-4