The structure of Bell inequalities

TL;DR

This paper analyzes the structure of Bell inequalities for N qubits, revealing their relation to Hadamard matrices and introducing Bell polynomials to generate and characterize these inequalities.

Contribution

It introduces a recursive method to generate Bell inequalities using Hadamard matrices and Bell polynomials, and characterizes their key features.

Findings

Most inequalities include all expectation values considered

Established recursive generation of inequalities from N=1

Connected Bell inequalities to Hadamard matrices

Abstract

We present an analysis of the structure of Bell inequalities, mainly for the case of N qubits with two observables each. We show that these inequalities are related to Hadamard matrices and define Bell polynomials (in one variable) as an additional tool. With these aids we raise several conditions the coefficients of Bell inequalities must satisfy, and recursively generate the whole set of inequalities starting from N=1. Moreover, we prove some characteristic features of this set, such as that most of the inequalities contain all expectation values under consideration. Finally, we show how the presented results can be used to construct Bell inequalities with certain properties. An outlook on further research topics concludes the paper.