Bigalois Extensions and the Graph Isomorphism Game
| Autor: | Mateusz Wasilewski, Kari Eifler, Michael Brannan, Samuel J. Harris, Xiaoyu Su, Alexandru Chirvasitu, Vern I. Paulsen |
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| Rok vydání: | 2019 |
| Předmět: |
Discrete mathematics
010102 general mathematics Mathematics - Operator Algebras FOS: Physical sciences Statistical and Nonlinear Physics Mathematical Physics (math-ph) Automorphism 01 natural sciences Quantum graph Mathematics - Quantum Algebra 0103 physical sciences FOS: Mathematics Quantum Algebra (math.QA) Graph homomorphism 010307 mathematical physics Compact quantum group Isomorphism 0101 mathematics Quantum information Graph isomorphism Operator Algebras (math.OA) Quantum Mathematical Physics Mathematics |
| Zdroj: | Communications in Mathematical Physics. 375:1777-1809 |
| ISSN: | 1432-0916 0010-3616 |
| Popis: | 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. Comment: 33 pages |
| Databáze: | OpenAIRE |
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