Bigalois Extensions and the Graph Isomorphism Game

We study the graph isomorphism game that arises in quantum information theory. We prove that the non-commutative algebraic notion of a quantum isomorphism between two graphs is same as the more physically motivated one arising from the existence of a perfect quantum strategy for graph isomorphism game. This is achieved by showing that every algebraic quantum isomorphism between a pair of (quantum) graphs X and Y arises from a certain measured bigalois extension for the quantum automorphism groups G_X and G_Y of 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 a compact quantum group G is monoidally equivalent to the quantum automorphism group G_X of a given quantum graph X if and only if G is the quantum automorphism group of a quantum graph that is algebraically quantum isomorphic to X. Using the notion of equivalence for non-local games, we apply our results to other synchronous games, including the synBCS game and certain related graph homomorphism games.