Violation of Bell inequality based on $S_4$ symmetry

TL;DR

This paper extends a finite group representation method to analyze Bell inequality violations using $S_4$ symmetry related to a tetrahedral geometry, demonstrating quantum violation and exploring the associated nonlocal game.

Contribution

It applies the $S_4$ symmetry group to Bell inequality analysis, revealing quantum violations in a more complex symmetry setting than previous studies.

Findings

Bell inequality is violated in quantum theory with $S_4$ symmetry.

The method relates Bell inequalities to tetrahedral symmetry.

Nonlocal game analysis confirms quantum violation.

Abstract

In two recent papers (Phys. Rev. A90 (2014), 062121 and Phys. Rev. A91 (2015), 052110) an interesting method of analyzing the violation 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 $S_4$ symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of threedimensional representation of $S_4$ 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.