Possible Vulnerability of Bell-Clauser-Horne-Shimony-Holt Tests used for Quantum Certification

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.