A hidden variables (HVs) model is reported, which reproduces quantum predictions for Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) tests. The existence of such a model poses some limitations to quantum certifications that rely on Bell-CHSH inequality violations. The reported model does not prove wrong Bell's theorem. The latter assumes the factorability of the probability density pAB, which rules the stochastic behavior of the HVs. The reported HVs model is based on an extended form of pAB, which is suggested by Lebesgue's decomposition theorem for bounded functions. The considered pAB complies with locality and realism, and also with measurement independence, parameter independence and outcome independence.
ABSTRACT A hidden variables (HVs) model is reported, which reproduces quantum predictions for Bell‐Clauser‐Horne‐Shimony‐Holt (Bell‐CHSH) tests. The existence of such a model poses some limitations to quantum certifications that rely on Bell‐CHSH inequality violations. The reported model does not prove wrong Bell's theorem. The latter assumes the factorability of the probability density p AB , which rules the stochastic behavior of the HVs. The reported HVs model is based on an extended form of p AB , which is suggested by Lebesgue's decomposition theorem for bounded functions. The considered p AB complies with locality and realism, and also with measurement independence, parameter independence and outcome independence.
In 1982, Alain Aspect, and collaborators performed an experiment, in order to observe the violation of the inequality of Clauser-Horne-Shimony-Holt. After the experiment, they used the data in the inequality and concluded that the inequality, obtained through probabilistic arguments, was violated confirming the conclusions obtained by John S. Bell that was not possible a theory of hidden variables in the conditions suggested by Einstein, Podolsky and Rosen. The aim of this work is to verify the validity of the inequalities in the conditions of the probabilistic theory, in order to hold with the axioms of the probability.