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International Journal of Theoretical Physics

Bell Inequalities and Group Symmetry

Katarzyna Bolonek-Lasoń

Published Mar 3, 2017• 0 citations
Publisher

Abstract

Recently the method based on irreducible representations of finite groups has been proposed as a tool for investigating the more sophisticated versions of Bell inequalities (V. Ugǔr Gűney, M. Hillery, Phys. Rev. A90, 062121 ([2014]) and Phys. Rev. A91, 052110 ([2015])). In the present paper an example based on the symmetry group S 4 is considered. The Bell inequality violation due to the symmetry properties of regular tetrahedron is described. A nonlocal game based on the inequalities derived is described and it is shown that the violation of Bell inequality implies that the quantum strategies outperform their classical counterparts.

DOI
10.1007/s10773-017-3323-9

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Breaking of Bell inequality based on S4S_4S4​ symmetry
In two recent papers (Phys. Rev. A90 (2014), 062121 and Phys. Rev. A91 (2015), 052110) an interesting method of analyzing the breaking of Bell inequalities has been proposed which is based on the theory of finite group representations. We apply here this method to more complicated example of S4S_4S4​ symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of threedimensional representation of S4S_4S4​ determined by the geometry of tetrahedron we find that the Bell inequality under consideration is broken in quantum theory. The corresponding nonlocal game is analyzed.
In two recent papers [Phys. Rev. A 90, 062121 (2014) and Phys. Rev. A, 91, 052110 (2015)] an interesting method of analyzing the violation of Bell inequalities has been proposed which is based on the theory of finite group representations. Here we apply this method to more complicated examples of S4{S}_{4}S4​ symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of three-dimensional representations of S4{S}_{4}S4​ determined by the geometry of tetrahedron we find that the Bell inequality under consideration is violated in quantum theory. The corresponding nonlocal game is analyzed.
Violation of Bell inequalities from S4S_4S4​ S 4 symmetry: the three orbits case
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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.
Bell inequalities for S4S_4S4​ group: classical and quantum bounds for two orbits case
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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 partite system, 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 ddd, 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. Moreover, all the corresponding nonlocal games are analyzed.