  At first sight, the use of an everywhere positive Wigner function as a
probability density to perform stochastic simulations in quantum optics seems
equivalent to the introduction of local hidden variables, thus preventing any
violation of Bell inequalities. However, because of the difference between
symmetrically and normally ordered operators, some trajectories in stochastic
simulations can imply negative intensities, despite a positive mean. Hence,
Bell inequalities do not apply. Here, we retrieve for a weakly squeezed
Gaussian state the maximum violation on polarization states allowed by quantum
mechanics, for the Clauser-Horn-Shimony-Holt (CHSH), as well as for the
Clauser-Horn Bell inequalities. For the case of the Clauser-Horn Bell
inequality, the influence of the quantum efficiency of the detectors is
studied, and for both inequalities, the influence of the degree of squeezing is
assessed, as well as the uncertainty range versus the number of trajectories
used in the simulations.
