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arXiv:2006.15537 · Physica Scripta

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

Éric Lantz, Mehdi Mabed, F. Devaux

Published Jan 20, 2021• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1402-4896/abdd56

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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.
The experimental test of Bell’s inequality is mainly focused on Clauser–Horne–Shimony–Holt (CHSH) form, which provides a quantitative bound, while little attention has been paid to the violation of Wigner inequality (WI). Based on the spin coherent state quantum probability statistics, we in the present paper extend the WI and its violation to arbitrary two-spin entangled states with antiparallel and parallel spin-polarizations. The local part of density operator gives rise to the WI while the violation is a direct result of nonlocal interference between two components of the entangled states. The Wigner measuring outcome correlation denoted by [Formula: see text] is always less than or at most equal to zero for the local realist model ([Formula: see text]) regardless of the specific initial state. On the other hand, the violation of WI is characterized by any positive value of [Formula: see text], which possesses a maximum violation bound [Formula: see text] [Formula: see text]. We conclude that the WI is equally convenient for the experimental test of violation by the quantum entanglement.
In this work we aim to analyze the Clauser-Horne-Shimony-Holt CHSH inequality strictly in the context of probability theory. In the course of assembling inequality we have to take care not to produce assumptions a priori, that is, physically or intuitively accepted suppositions. Of course, this does not mean that after these considerations, we put the adequate physical conditions suitable and generally accepted in these contexts. This allows us to clearly visualize the possibility of finding a greater inequality than that of CHSH in which it is included. This inequality does not contradict the CHSH inequality. This result is suported by using a robust computational simulation, showing the possibility for obtaining an inequality for quantum mechanics that is not violated and allows random hidden variables without any conflict with Bell's inequality.
We discuss the relationship between the Bogoliubov transformations, squeezed states, entanglement, and maximum violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. In particular, we point out that the construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in a simple and general way, covering a large variety of models, ranging from quantum mechanism to relativistic quantum field theories. Various examples are employed to illustrate the above mentioned framework. We start by considering a pair of entangled spin-one particles and a squeezed oscillator in quantum mechanics, moving then to the relativistic complex quantum scalar field and to the analysis of the vacuum state in Minkowski space-time in terms of the left and right Rindler modes. In the latter case, the Bell-CHSH inequality turns out to be parametrized by the Unruh temperature.
In 1982, Alain Aspect, and collaborators performed an experiment, in order to observe the violation of the inequality of Clauser-Horne-Shimony-Holt. After the experiment, they used the data in the inequality and concluded that the inequality, obtained through probabilistic arguments, was violated confirming the conclusions obtained by John S. Bell that was not possible a theory of hidden variables in the conditions suggested by Einstein, Podolsky and Rosen. The aim of this work is to verify the validity of the inequalities in the conditions of the probabilistic theory, in order to hold with the axioms of the probability.
Bell's theorem shows that no local hidden-variable model can explain the measurement statistics of a quantum system shared between two parties, thus ruling out a classical (local) understanding of nature. In this paper we demonstrate that by relaxing the positivity restriction in the hidden-variable probability distribution it is possible to derive quasiprobabilistic Bell inequalities whose sharp upper bound is written in terms of a negativity witness of said distribution. This provides an analytic solution for the amount of negativity necessary to violate the Clauser-Horne-Shimony-Holt inequality by an arbitrary amount, therefore revealing the amount of negativity required to emulate the quantum statistics in a Bell test.
We investigate the role played by quantum operator ordering in the correlations that characterize two-photon polarization Bell measurements. The Clauser-Horne-Shimony-Holt (CHSH) criterion is investigated in the normal ordering imposed by the photodetection theory and in the symmetric ordering that constitutes the standard prescription for building Hermitian operators from products of noncommuting observables. The two approaches are obtained in a single theoretical framework, where operator ordering is directly associated with the representation used for the density matrix. Moreover, this discussion can be recast in terms of the contribution given by the vacuum fluctuations to the detected signals. We also envisage possible detection schemes sensitive to these fluctuations with recent technological developments.