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

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.