Operator solutions of linear systems and small cancellation

TL;DR

This paper demonstrates that certain graph-based linear systems admit quantum solutions and perfect strategies in nonlocal games, highlighting differences between quantum and classical approaches.

Contribution

It introduces conditions under which the incidence systems of graphs have quantum solutions over finite fields, revealing new quantum-classical separations in nonlocal games.

Findings

Existence of quantum solutions for graphs with specific degree and girth

Construction of linear systems with perfect quantum but no classical strategies

Identification of graph parameters influencing quantum solutions

Abstract

We show that if a graph has minimum vertex degree at least d and girth at least g, where (d, g) is (3, 6) or (4, 4), then the incidence system of the graph has a (possibly infinite-dimensional) quantum solution over $\mathbb{Z}_p$ for every choice of vertex weights and integer $p \geq 2$. In particular, there are linear systems over $\mathbb{Z}_p$, for $p$ an odd prime, such that the corresponding linear system nonlocal game has a perfect commuting-operator strategy, but no perfect classical strategy.