Intrinsic Mirror Symmetry and Robustness of Optimal Nonlocal Operators in One-Dimensional Quantum Spin Chains

TL;DR

This paper investigates the intrinsic mirror symmetry and robustness of optimal nonlocal operators in 1D quantum spin chains, revealing stable measurement structures that simplify experimental Bell tests.

Contribution

It uncovers the mirror symmetry of optimal single-site operators and demonstrates their structural stability across different quantum phases in translationally invariant chains.

Findings

Optimal single-site operator $\u0301p$ has intrinsic mirror symmetry.

Optimal nonlocal operator $\u0301S(\u0301p)$ remains structurally stable across quantum phases.

Results simplify experimental implementation of multipartite nonlocality tests.

Abstract

Multipartite nonlocality has been extensively investigated within one-dimensional quantum lattices. Previous research has primarily focused on the nonlocality measure $S$, which quantifies the violation of Bell-type inequalities. However, the optimal nonlocal operators, which are related to specific experimental settings required to achieve the violation, often remain elusive. In this work, we employ a string-like nonlocal operator $\hat{S}_N$, characterized by a core single-site operator $\hat{p}$, to investigate the optimal measurement setting in translationally invariant quantum chains. By analyzing the infinite-size transverse-field Ising, Cluster-Ising, and extended Ising models, we uncover two general results. First, for typical ground states, we find that the optimal single-site operator $\hat{p}$ possesses an intrinsic mirror symmetry. Second, the optimal nonlocal operator $\hat{S}(\hat{p})$ exhibits remarkable robustness: for a specific model, as the Hamiltonian parameter changes, the structure of $\hat{p}$ remains stable and persists across distinct quantum phases. These findings not only redefine the numerical optimization paradigm for multipartite nonlocality, but also significantly simplify the experimental requirements by identifying fixed measurement bases. This structural stability provides practical guidance for implementing macroscopic Bell tests in large-scale quantum simulators, making it highly compatible with modern efficient measurement protocols.