Optimizing quantum violation for multipartite facet Bell inequalities

TL;DR

This paper introduces a gradient-based optimization method to find tight Bell inequalities with maximal quantum violations in multipartite systems, aiding the understanding and experimental certification of quantum nonlocality.

Contribution

A novel gradient-based approach for optimizing Bell inequalities, enabling the discovery of facet inequalities with large quantum violations in complex multipartite scenarios.

Findings

Local maxima often correspond to facet Bell inequalities.

Method finds tight Bell inequalities with large quantum violations.

Analytical maximum ratio derived in the thermodynamic limit.

Abstract

Nonlocality shapes quantum correlations, revealed through the violation of Bell inequalities. The intersection of all valid Bell inequalities is the so-called local polytope. In multipartite systems, characterizing the local polytope quickly becomes an intractable task as the system size increases. Optimizing Bell inequalities to maximize the ratio between their quantum value and classical bound is key to understanding multipartite nonlocality. We propose a gradient-based method for this optimization. Numerical results indicate that local maxima of this ratio typically correspond to facet Bell inequalities of the local polytope. This enables an iterative search for tight and robust Bell inequalities. Applied to permutation-invariant scenarios, the method provides tight Bell inequalities with large quantum violations and facilitates experimental certification of Bell correlations without full knowledge of the local polytope. Moreover, analytical results of the maximum ratio are derived in the thermodynamic limit.