Robust violation of a multipartite Bell inequality from the perspective of a single-system game

TL;DR

This paper explains the robustness of violations of a generalized Bell inequality in multipartite quantum systems by mapping it to a single-qubit game, revealing symmetry and degeneracy as key factors.

Contribution

It provides a novel mapping from a multipartite Bell inequality to a single-qubit game, elucidating the origin of robust violations through symmetry and degeneracy.

Findings

Degeneracy of 2^{N-2} in the generalized CHSH operators.

Robust violations are linked to symmetry of maximally entangled states.

Mapping explains robustness under specific noise conditions.

Abstract

Recently, Fan \textit{et al.} [Mod. Phys. Lett. A 36, 2150223 (2021)], presented a generalized Clauser-Horne-Shimony-Holt (CHSH) inequality, to identify $N$-qubit Greenberger-Horne-Zeilinger (GHZ) states. They showed an interesting phenomenon that the maximal violation of the generalized CHSH inequality is robust under some specific noises. In this work, we map the inequality to the CHSH game, and consequently to the CHSH* game in a single-qubit system. This mapping provides an explanation for the robust violations in $N$-qubit systems. Namely, the robust violations, resulting from the degeneracy of the generalized CHSH operators correspond to the symmetry of the maximally entangled two-qubit states and the identity transformation in the single-qubit game. This explanation enables us to exactly demonstrate that the degeneracy is $2^{N-2}$.