Robust self-testing with CHSH mod 3

TL;DR

This paper determines the maximal quantum violation of the CHSH mod 3 Bell inequality and establishes its self-testing properties, showing near-optimal strategies are close to the unique optimal strategies.

Contribution

It precisely computes the maximal quantum value of the CHSH mod 3 inequality and proves its self-testing property with a quantitative robustness bound.

Findings

Maximal quantum violation of CHSH mod 3 is exactly determined.

There are four symmetry-related optimal strategies using maximally entangled two-qutrit states.

Any near-optimal strategy is close to an optimal irreducible strategy within an $O( oot extstyle rac{ ext{epsilon}}{})$ bound.

Abstract

The CHSH mod 3 Bell inequality is a natural testbed for higher-dimensional quantum nonlocality, yet its maximal quantum violation and self-testing properties have remained unresolved. We determine its exact maximal quantum value and show that, up to unitary equivalence and the natural symmetries of the inequality, it admits a unique optimal irreducible strategy; equivalently, there are four symmetry-related optimal irreducible strategies. Each of these strategies uses a maximally entangled two-qutrit state. We further prove that any strategy whose value is within $\varepsilon$ of the optimum is $O(\sqrt{\varepsilon})$-close, up to local isometries, to a direct sum of optimal irreducible strategies.