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

TL;DR

This paper demonstrates that stochastic simulations based on positive Wigner functions can violate Bell inequalities due to negative intensities in trajectories, revealing quantum nonlocality even with classical-like probability distributions.

Contribution

It shows that Gaussian states simulated with positive Wigner functions can violate Bell inequalities, highlighting the role of operator ordering and trajectory negativity in quantum nonlocality.

Findings

Maximum Bell inequality violation for weakly squeezed Gaussian states

Influence of detector efficiency on Bell inequality violations

Effect of squeezing degree and number of trajectories on results

Abstract

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.