Robust self-testing for linear constraint system games

TL;DR

This paper proves robust self-testing for certain linear constraint system games over modular rings, extending representation theory to nonabelian groups and providing bounds for specific quantum games.

Contribution

It extends the representation-theoretic framework to linear constraint games over , enabling robust self-testing bounds for Magic Square and Pentagram games over .

Findings

Proves robust self-testing for specific LCS games over .

Extends representation theory to nonabelian finite groups in quantum game analysis.

Shows impossibility of self-testing maximally entangled states of non-power-of-two dimensions.

Abstract

We study linear constraint system (LCS) games over the ring of arithmetic modulo $d$. We give a new proof that certain LCS games (the Mermin--Peres Magic Square and Magic Pentagram over binary alphabets, together with parallel repetitions of these) have unique winning strategies, where the uniqueness is robust to small perturbations. In order to prove our result, we extend the representation-theoretic framework of Cleve, Liu, and Slofstra (Journal of Mathematical Physics 58.1 (2017): 012202.) to apply to linear constraint games over $\mathbb{Z}_d$ for $d\geq 2$. We package our main argument into machinery which applies to any nonabelian finite group with a ''solution group'' presentation. We equip the $n$-qubit Pauli group for $n\geq 2$ with such a presentation; our machinery produces the Magic Square and Pentagram games from the presentation and provides robust self-testing bounds. The question of whether there exist LCS games self-testing maximally entangled states of local dimension not a power of $2$ is left open. A previous version of this paper falsely claimed to show self-testing results for a certain generalization of the Magic Square and Pentagram mod $d\neq 2$. We show instead that such a result is impossible.