Matroid isomorphism games
Daniel Corey, Simon Schmidt, Marcel Wack

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.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topology and Set Theory · Polynomial and algebraic computation
