# Bigalois extensions and the graph isomorphism game

## TL;DR

This paper explores the connection between quantum isomorphisms of graphs and bigalois extensions of quantum groups, establishing a correspondence that unifies various notions of quantum graph isomorphism and non-local games.

## Contribution

It demonstrates that algebraic quantum isomorphisms are quotients of bigalois extensions, showing monoidal equivalence of quantum automorphism groups and unifying different quantum isomorphism concepts.

## Key findings

- Quantum isomorphisms correspond to quotients of bigalois extensions.
- Quantum automorphism groups of isomorphic graphs are monoidally equivalent.
- Different notions of quantum graph isomorphism coincide for classical graphs.

## Abstract

We study the graph isomorphism game that arises in quantum information theory from the perspective of bigalois extensions of compact quantum groups. We show that every algebraic quantum isomorphism between a pair of (quantum) graphs $X$ and $Y$ arises as a quotient of a certain measured bigalois extension for the quantum automorphism groups $G_X$ and $G_Y$ of the graphs $X$ and $Y$. In particular, this implies that the quantum groups $G_X$ and $G_Y$ are monoidally equivalent. We also establish a converse to this result, which says that every compact quantum group $G$ monoidally equivalent to $G_X$ is of the form $G_Y$ for a suitably chosen quantum graph $Y$ that is quantum isomorphic to $X$. As an application of these results, we deduce that the $\ast$-algebraic, C$^\ast$-algebraic, and quantum commuting (qc) notions of a quantum isomorphism between classical graphs $X$ and $Y$ all coincide. Using the notion of equivalence for non-local games, we deduce the same result for other synchronous non-local games, including the synBCS game and certain related graph homomorphism games.