Effects of Topological Boundary Conditions on Bell Nonlocality

TL;DR

This paper explores how boundary conditions in 2D lattice systems influence the ability of Bell inequalities to detect nonlocal correlations, providing insights for certifying nonlocality in many-qubit devices.

Contribution

It establishes a link between lattice topology induced by boundary conditions and the effectiveness of Bell inequalities in revealing nonlocality, using tropical algebra and tensor networks.

Findings

Boundary conditions affect Bell inequality effectiveness

Topology influences nonlocality detection capacity

Guides certification of Bell nonlocality in quantum devices

Abstract

Bell nonlocality is the resource that enables device-independent quantum information processing tasks. It is revealed through the violation of so-called Bell inequalities, indicating that the observed correlations cannot be reproduced by any local hidden variable model. While well explored in few-body settings, the question of which Bell inequalities are best suited for a given task remains quite open in the many-body scenario. One natural approach is to assign Bell inequalities to physical Hamiltonians, mapping their interaction graph to two-body, nearest-neighbor terms. Here, we investigate the effect of boundary conditions in a two-dimensional square lattice, which can induce different topologies in lattice systems. We find a relation between the induced topology and the Bell inequality's effectiveness in revealing nonlocal correlations. By using a combination of tropical algebra and tensor networks, we quantify their detection capacity for nonlocality. Our work can act as a guide to certify Bell nonlocality in many-qubit devices by choosing a suitable Hamiltonian and measuring its ground state energy; a task that many quantum experiments are purposely built for.