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arXiv:quant-ph/0010091

Local hidden-variable models and negative-probability measures

Jose Luis Cereceda

Published Oct 26, 2000• 10 citations
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Abstract

Elaborating on a previous work by Han et al., we give a general, basis-independent proof of the necessity of negative probability measures in order for a class of local hidden-variable (LHV) models to violate the Bell-CHSH inequality. Moreover, we obtain general solutions for LHV-induced probability measures that reproduce any consistent set of probabilities.

DOI
10.48550/arxiv.quant-ph/0010091

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Elaborating on a previous work by Han et al., we give a general, basis-independent proof of the necessity of negative probability measures in order for a class of local hidden-variable (LHV) models to violate the Bell-CHSH inequality. Moreover, we obtain general solutions for LHV-induced probability measures that reproduce any consistent set of probabilities.
Geurdes (2014) outlines a probabilistic construction of a counterexample to Bell’s theorem. He gives a procedure to repeatedly sample from a specially constructed “pool” of local hidden variable models (depending on a table of numerically calculated parameters) and select from the results one LHV model, determining a random value S of the usual CHSH combination S of four (theoretical) correlation values. He claims Prob ( | S | > 2 ) > 0 . We expose a fatal flaw in the analysis: the procedure generates a non-local hidden variable model. To disprove this claim, Geurdes should program his procedure and generate random LHV’s till he finds one violating the CHSH inequality.
The essence of Bell’s theorem is that, in general, quantum statistics cannot be reproduced by a local hidden variable (LHV) model. This impossibility is strongly manifested when statistics collected by measuring certain local observables on a singlet state, violates the Bell inequality. In this work, we search for local POVMs with binary outcomes for which an LHV model can be constructed for a singlet state. We provide various subsets of observables for which an LHV model can be provided for singlet statistics.
Several authors have recently claimed that Bell's inequalities (BI) do not apply to certain types of generalized local hidden variables (HV) models. These claims are rejected, by means of a proof of BI valid for a very broad class of local HV models (deterministic or stochastic, with or without memory, time dependent or time independent, setting dependent or setting independent, and even allowing classical communication between the wings of the experiment). The precise meaning of the locality requirement is clarified during the proof. The exact roles of the two assumptions that amount to Bell's locality (parameter independence and outcome independence) are explicitly shown, using each one of them in a different step of the proof. using each one of them in a different step of the proof.
Violation of the CHSH inequality supposedly demonstrates an irreconcilable conflict between quantum mechanics and local, realistic hidden variable theories. We show that the mathematical assumptions underlying the proof of the CHSH inequality are, in fact, incompatible with the physics of the experiments testing such inequality. This implies that we cannot dismiss local realistic hidden variable theories on the basis of currently available experimental data yet. However, we also show that an experimental proof of CHSH inequality is, in principle, possible, but it is unclear how to implement, in practice, such an experiment.
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.