Groups, Platonic solids and Bell inequalities

Katarzyna Bolonek-Lasoń,Piotr Kosiński

doi:10.22331/q-2021-11-29-593

Abstract

The construction of Bell inequalities based on Platonic and Archimedean solids (Quantum 4 (2020), 293) is generalized to the case of orbits generated by the action of some finite groups. A number of examples with considerable violation of Bell inequalities is presented.

Highlights

Since the pioneering Bell paper [1] the Bell inequalities became the subject of intensive study. Their importance stems from the fact that their violation at the quantum level provides the evidence that the quantum theory cannot be viewed as a local realistic theory. Another important notion in physics is that of symmetry
The sets {vi}Ni=A1, {wj}Nj=B1 of Alice and Bob settings are the orbits of the symmetry group generated by the action of the representation matrices on some carefully selected initial vectors
We have shown here that one can take as a starting point the symmetry groups of Platonic solids or, more precisely, the groups generating these solids as orbits of their threedimensional real representations

Summary

Introduction

Since the pioneering Bell paper [1] the Bell inequalities became the subject of intensive study (for a review see [2], [3]). It appears natural to study Bell inequalities for the systems described by the sets of states classified by the representations of some groups This idea has been proposed in the interesting papers by Güney and Hillery [4], [5] and studied in some detail by the present authors [6,7,8,9]. In the subsequent paper we will analyze the Bell inequalities for the Alice and Bob settings exhibiting ”hidden” symmetry, i.e. generated by group action but not coinciding with Platonic or Archimedean solids

The general scheme

Some examples

Summary

A Appendix

B Appendix