Characterizing Translation-Invariant Bell Inequalities using Tropical Algebra and Graph Polytopes

TL;DR

This paper introduces a novel approach using tropical algebra and graph theory to characterize translation-invariant Bell inequalities in one-dimensional quantum systems, providing efficient methods to analyze their extremal points and tightness.

Contribution

It develops a new methodology combining tropical algebra tensor networks and graph theory to analyze translation-invariant Bell inequalities, extending previous work in the field.

Findings

TI Bell polytope has a bounded number of extremal points

Efficient method to list all vertices for specific system sizes

Reinterpretation of tropical algebra and graph concepts in Bell nonlocality

Abstract

Nonlocality is one of the key features of quantum physics, which is revealed through the violation of a Bell inequality. In large multipartite systems, nonlocality characterization quickly becomes a challenging task. A common practice is to make use of symmetries, low-order correlators, or exploiting local geometries, to restrict the class of inequalities. In this paper, we characterize translation-invariant (TI) Bell inequalities with finite-range correlators in one-dimensional geometries. We introduce a novel methodology based on tropical algebra tensor networks and highlight its connection to graph theory. Surprisingly, we find that the TI Bell polytope has a number of extremal points that can be uniformly upper-bounded with respect to the system size. We give an efficient method to list all vertices of the polytope for a particular system size, and characterize the tightness of a given TI Bell inequality. The connections highlighted in our work allow us to re-interpret concepts developed in the fields of tropical algebra and graph theory in the context of Bell nonlocality, and vice-versa. This work extends a parallel article [M. Hu \textit{et al.}, arXiv: 2208.02798 (2022)] on the same subject.