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arXiv:1907.10437

Bell inequalities for $S_4$ group: classical and quantum bounds for two orbits case

Katarzyna Bolonek-Lasoń, Piotr Kosiński

Published Jul 24, 2019• 0 citations
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Abstract

Within Guney-Hillery approach a number of examples of classical and quantum bounds on sum of probabilities resulting from two orbits of $S_4$ is considered. It is shown that the violation of Bell's inequalities is rather rare and gentle.

DOI
10.48550/arxiv.1907.10437

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Violation of Bell inequalities from S4S_4S4​ S 4 symmetry: the three orbits case
The recently proposed (Güney and Hillery in Phys Rev A 90:062121, 2014; Phys Rev A 91:052110, 2015) group theoretical approach to the problem of violating the Bell inequalities is applied to S4S_4S4​ group. The Bell inequalities based on the choice of three orbits in the representation space corresponding to standard representation of S4S_4S4​ are derived and their violation is described. The corresponding nonlocal games are analyzed.
Bell inequalities for S4S_4S4​ group: classical and quantum bounds for two-orbit case
In recent papers, the theory of representations of finite groups has been proposed to analyzing the violation of Bell inequalities. In this paper, we apply this method to more complicated cases. For two parties, Alice and Bob each make one of ddd possible measurements, each measurement has nnn outcomes, the Bell inequalities based on the choice of two orbits are derived. The classical bound is only dependent on the number of measurements, but the quantum bound is dependent both on nnn and ddd. Even so, when ddd is large enough, the quantum bound is only dependent on ddd. The subset of probabilities for four parties based on the choice of six orbits under group action is derived and its violation is described. Restricting the six orbits to three parties by forgetting the last party, and guaranteeing the classical bound invariant, the Bell inequality based on the choice of four orbits is derived. All the corresponding nonlocal games are analyzed.
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