Bigalois Extensions and the Graph Isomorphism Game

Michael Brannan, Alexandru Chirvasitu, Kari Eifler, Samuel Harris, Vern Paulsen, Xiaoyu Su, Mateusz Wasilewski

Research output: Contribution to journalArticlepeer-review

29 Scopus citations

Abstract

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 GX and GY of X and Y. In particular, this implies that the quantum groups GX and GY 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 GX 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.

Original languageEnglish (US)
Pages (from-to)1777-1809
Number of pages33
JournalCommunications in Mathematical Physics
Volume375
Issue number3
DOIs
StatePublished - May 1 2020
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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