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At first sight, the use of an everywhere positive Wigner function as aprobability density to perform stochastic simulations in quantum optics seemsequivalent to the introduction of local hidden variables, thus preventing anyviolation of Bell inequalities. However, because of the difference betweensymmetrically and normally ordered operators, some trajectories in stochasticsimulations can imply negative intensities, despite a positive mean. Hence,Bell inequalities do not apply. Here, we retrieve for a weakly squeezedGaussian state the maximum violation on polarization states allowed by quantummechanics, for the Clauser-Horn-Shimony-Holt (CHSH), as well as for theClauser-Horn Bell inequalities. For the case of the Clauser-Horn Bellinequality, the influence of the quantum efficiency of the detectors isstudied, and for both inequalities, the influence of the degree of squeezing isassessed, as well as the uncertainty range versus the number of trajectoriesused in the simulations.

Violation of Bell inequalities by stochastic simulations of Gaussian States based on their positive Wigner representation