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.

乍一看，在量子光学中使用随处可见的正Wigner函数作为概率密度来执行随机模拟似乎等同于引入局部隐藏变量，从而避免了任何违反Bell不等式的情况。但是，由于对称算子和通常有序算子之间的差异，尽管均值是正数，但是随机模拟中的某些轨迹可能暗示负强度。因此，贝尔不等式不适用。在这里，我们针对弱压缩的高斯状态检索了量子力学，Clauser-Horn-Shimony-Holt（CHSH）以及Clauser-Horn Bell不等式允许的量子力学所允许的最大极化态违规。对于Clauser-Horn Bell不等式，研究了探测器量子效率的影响，并且对于这两个不等式，