Nonlocality, Integrability and Quantum Chaos in the Spectrum of Bell Operators

TL;DR

This paper explores the spectral properties of a Bell operator in many-body three-level systems, revealing a surprising link between nonlocality, integrability, and quantum chaos through spectral statistics analysis.

Contribution

It introduces a permutationally invariant Bell inequality for many-body systems and connects Bell nonlocality with spectral integrability and chaos in the associated Bell operator.

Findings

Maximal Bell violation correlates with Poissonian (integrable) spectral statistics.

Generic perturbations lead to Wigner-Dyson (chaotic) spectral behavior.

An emergent parity symmetry explains spectral regularity near maximal violation.

Abstract

We introduce a permutationally invariant multipartite Bell inequality for many-body three-level systems and use it to investigate a connection between Bell nonlocality and (lack of) quantum chaos. An associated Bell operator is then defined via Born's rule, mapping the conditional probabilities of the Bell inequality to quantum measurement operators. This allows us to interpret the Bell operator as an effective Hamiltonian, which we use to analyze its spectral statistics across different SU(3) irreducible representations and measurement choices. Surprisingly, we find that, in every irreducible representation exhibiting nonlocality, the measurement settings yielding maximal violation result in a Bell operator with Poissonian level statistics, thus signaling integrable behavior. This integrability is both unique and fragile, since generic or slightly perturbed measurements lead to the Wigner-Dyson statistics associated with chaotic behavior. Through further analysis, we are able to identify an emergent parity symmetry in the Bell operator near the point of maximal violation, providing an explanation for the observed regularity in the spectrum. These results suggest a deep interplay between optimal quantum measurements, non-local correlations, and integrability, opening new perspectives at the intersection of Bell nonlocality and quantum chaos.