The Morita theory of quantum graph isomorphisms

TL;DR

This paper classifies quantum graph isomorphisms using Morita theory and quantum automorphism groups, revealing a deep connection between quantum symmetries and classical automorphism subgroups.

Contribution

It introduces a Morita-theoretic framework for understanding quantum graph isomorphisms via Frobenius algebras and automorphism subgroups, providing a new classification method.

Findings

Quantum isomorphic graphs correspond to Morita equivalence classes of Frobenius algebras.

Graphs with no quantum symmetry are classified by certain automorphism subgroups.

Almost all large graphs with no quantum symmetry are not quantum isomorphic to non-isomorphic graphs.

Abstract

We classify instances of quantum pseudo-telepathy in the graph isomorphism game, exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups. Specifically, we show that graphs quantum isomorphic to a given graph are in bijective correspondence with Morita equivalence classes of certain Frobenius algebras in the category of finite-dimensional representations of the quantum automorphism algebra of that graph. We show that such a Frobenius algebra may be constructed from a central type subgroup of the classical automorphism group, whose action on the graph has coisotropic vertex stabilisers. In particular, if the original graph has no quantum symmetries, quantum isomorphic graphs are classified by such subgroups. We show that all quantum isomorphic graph pairs corresponding to a well-known family of binary constraint systems arise from this group-theoretical construction. We use our classification to show that, of the small order vertex-transitive graphs with no quantum symmetry, none is quantum isomorphic to a non-isomorphic graph. We show that this is in fact asymptotically almost surely true of all graphs.