Matroid isomorphism games

TL;DR

This paper introduces matroid isomorphism games to characterize matroid isomorphism, explores quantum isomorphism with nonisomorphic matroids, and defines a quantum automorphism group, revealing nonclassical symmetries.

Contribution

It develops a framework connecting matroid isomorphism with nonlocal games and introduces quantum isomorphism, including algebraic and automorphism group characterizations.

Findings

Existence of nonisomorphic quantum isomorphic matroids

A new algebraic characterization of quantum isomorphism

Definition of a quantum automorphism group for matroids

Abstract

We define and study a collection of matroid isomorphism games corresponding to various axiomatic characterizations of matroids. These are nonlocal games played between two cooperative players. Each game is played on two matroids, and the matroids are isomorphic if and only if the game has a perfect classical winning strategy. We define notions of quantum isomorphism in terms of perfect quantum commuting strategies, and we find a pair of nonisomorphic matroids that are quantum isomorphic. We also give a purely algebraic characterization of quantum isomorphic matroids. Finally, we use this notion of quantum isomorphism to describe a new type of quantum automorphism group of a matroid and derive a sufficient condition for a matroid to have nonclassical quantum automorphism.