The hypergraph isomorphism game, Hopf algebras and Galois extensions

TL;DR

This paper develops an algebraic framework for quantum hypergraph isomorphisms using compact quantum groups, revealing new distinctions from classical isomorphisms and linking algebraic and operational perspectives.

Contribution

It introduces a synchronous hypergraph isomorphism game, connects quantum isomorphisms with bi-Galois extensions, and characterizes quantum isomorphisms through algebraic structures.

Findings

Existence of hypergraphs that are quantum isomorphic but not classically isomorphic

The hypergraph game algebra is a quotient of the graph isomorphism game algebra

Hypergraph game algebra forms a bi-Galois extension over quantum automorphism groups

Abstract

We develop an algebraic and operational framework for quantum isomorphisms of hypergraphs, using tools from compact quantum group theory. We introduce a new synchronous version of the hypergraph isomorphism game whose game algebra uniformly encodes multiple notions of quantum isomorphisms of hypergraphs. We show that there exist hypergraphs that are quantum isomorphic but not classically isomorphic. For graphs, we show that the $*$-algebra of the hypergraph isomorphism game is a quotient of the $*$-algebra of the graph isomorphism game. We further prove that the hypergraph game algebra forms a bi-Galois extension over the quantum automorphism groups of the underlying hypergraphs. This allows us to deduce that the algebraic notion of a quantum isomorphism of hypergraphs coincides with the operational one coming from the existence of perfect quantum strategies. Viewing games themselves as hypergraphs, we analyze isomorphisms and the transfer of strategies within this setting. Finally, we construct a $*$-algebra whose representation theory characterizes distinct classes of quantum isomorphisms between non-local games.