Bell inequalities from group actions of single-generator groups

TL;DR

This paper introduces a method to generate Bell inequalities using group actions of single-generator abelian groups, linking group theory with quantum nonlocality and nonlocal games.

Contribution

It presents a novel approach to deriving Bell inequalities through group actions, expanding the tools for analyzing quantum nonlocality.

Findings

Derived Bell inequalities for various measurement settings

Connected group actions to nonlocal game formulations

Demonstrated violations of inequalities with quantum states

Abstract

We study a method of generating Bell inequalities by using group actions of single-generator abelian groups. Two parties, Alice and Bob, each make one of M possible measurements on a system, with each measurement having K possible outcomes. The probabilities for the outcomes of these measurements are P(a_j = k, b_{j'}=k'), where j,j' are in the set {1,2,... M} and k,k' are in the set {0,1,... K-1}. The sums of some subsets of these probabilities have upper bounds when the probabilities result from a local, realistic theory that can be violated if the probabilities come from quantum mechanics. In our case the subsets of probabilities are generated by a group action, in particular, a representation of a single-generator group acting on product states in a tensor-product Hilbert space. We show how this works for several cases, including M=2, K=3, and general M, K=2. We also discuss the resulting inequalities in terms of nonlocal games.