2018/12/31 by Michael Brannan, Alexandru Chirvasitu, Alexandru Chirvăsitu +7 · 46 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics #Discrete mathematics #Graph #Graph homomorphism #Graph isomorphism #Homomorphism #Isomorphism (crystallography) #Isomorphism extension theorem #Line graph #Mathematics #Physics #Pure mathematics #Quantum #Quantum mechanics #Quotient #Voltage graph #math-ph #math.MP #math.OA #math.QA
paper · pdf · open access · doi:10.1007/s00220-019-03563-9
published in Communications in Mathematical Physics 375(3), 1777-1809 (Springer Science+Business Media) · 33 pages
openalex publication_date 2019/09/14 · arxiv created 2020/11/03 · arxiv updated 2020/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 GX and GY of the graphs 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 every compact quantum group G monoidally equivalent to GX is of the form GY for a suitably chosen quantum graph Y that is quantum isomorphic to X. As an application of these results, we deduce that the ∗-algebraic, C^∗-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.