Arkhipov's theorem, graph minors, and linear system nonlocal games

TL;DR

This paper extends Arkhipov's theorem on graph incidence games, linking perfect quantum strategies to graph minors and group properties, and introduces forbidden minor characterizations for properties like finiteness and abelianness.

Contribution

It generalizes Arkhipov's theorem by characterizing quotient closed properties of solution groups via forbidden minors in two-coloured graphs.

Findings

Arkhipov's theorem rederived from a group theoretic perspective.

Forbidden minors identified for properties like finiteness and abelianness.

Methods are purely combinatorial, enabling further property characterizations.

Abstract

The perfect quantum strategies of a linear system game correspond to certain representations of its solution group. We study the solution groups of graph incidence games, which are linear system games in which the underlying linear system is the incidence system of a (non-properly) two-coloured graph. While it is undecidable to determine whether a general linear system game has a perfect quantum strategy, for graph incidence games this problem is solved by Arkhipov's theorem, which states that the graph incidence game of a connected graph has a perfect quantum strategy if and only if it either has a perfect classical strategy, or the graph is nonplanar. Arkhipov's criterion can be rephrased as a forbidden minor condition on connected two-coloured graphs. We extend Arkhipov's theorem by showing that, for graph incidence games of connected two-coloured graphs, every quotient closed property of the solution group has a forbidden minor characterization. We rederive Arkhipov's theorem from the group theoretic point of view, and then find the forbidden minors for two new properties: finiteness and abelianness. Our methods are entirely combinatorial, and finding the forbidden minors for other quotient closed properties seems to be an interesting combinatorial problem.