Generalizing optimal Bell inequalities

TL;DR

This paper introduces a method to systematically generalize Bell inequalities to more particles, enabling better detection of quantum nonlocality, especially for states not identifiable by existing inequalities.

Contribution

A novel approach to characterize and generalize Bell inequalities under linear constraints, facilitating the discovery of new inequalities for multiple particles.

Findings

All generalizations of the Froissart I3322 inequality to three particles were identified.

Some generalized inequalities can detect nonlocality where two-setting inequalities cannot.

The method enables systematic exploration of Bell inequalities with symmetry or other linear constraints.

Abstract

Bell inequalities are central tools for studying nonlocal correlations and their applications in quantum information processing. Identifying inequalities for many particles or measurements is, however, difficult due to the computational complexity of characterizing the set of local correlations. We develop a method to characterize Bell inequalities under constraints, which may be given by symmetry or other linear conditions. This allows to search systematically for generalizations of given Bell inequalities to more parties. As an example, we find all possible generalizations of the two-particle inequality by Froissart [Il Nuovo Cimento B64, 241 (1981)], also known as I3322 inequality, to three particles. For the simplest of these inequalities, we study their quantum mechanical properties and demonstrate that they are relevant, in the sense that they detect nonlocality of quantum states, for which all two-setting inequalities fail to do so.