Tropical contraction of tensor networks as a Bell inequality optimization toolset

TL;DR

This paper introduces a novel tensor network contraction method in tropical algebra to efficiently compute classical bounds of Bell inequalities, applicable to multipartite and bipartite systems, including the thermodynamic limit.

Contribution

It presents a new tropical algebra-based framework for Bell inequality optimization, linking tropical eigenvalues with classical bounds and extending to large-scale systems.

Findings

Tropical tensor network contraction effectively computes Bell bounds.

Method extends to thermodynamic limit for translationally invariant systems.

Connection established between tropical eigenvalues and classical bounds per particle.

Abstract

We show that finding the classical bound of broad families of Bell inequalities can be naturally framed as the contraction of an associated tensor network, but in tropical algebra, where the sum is replaced by the minimum and the product is replaced by the arithmetic addition. We illustrate our method with paradigmatic examples both in the multipartite scenario and the bipartite scenario with multiple outcomes. We showcase how the method extends into the thermodynamic limit for some translationally invariant systems and establish a connection between the notions of tropical eigenvalue and the classical bound per particle as a fixed point of a tropical renormalization procedure.