2025/10/21 by Georgios Baziotis, Baziotis, Georgios, Alexandros Chatzinikolaou +3
Computer Science · Mathematics · #16T20 #20G42 #46L52 #81P45 #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2510.18679
openalex publication_date 2025/10/21 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We develop an algebraic and operational framework for quantum isomorphisms of hypergraphs, using tools from compact quantum group theory. We introduce a new synchronous version of the hypergraph isomorphism game whose game algebra uniformly encodes multiple notions of quantum isomorphisms of hypergraphs. We show that there exist hypergraphs that are quantum isomorphic but not classically isomorphic. For graphs, we show that the *-algebra of the hypergraph isomorphism game is a quotient of the *-algebra of the graph isomorphism game. We further prove that the hypergraph game algebra forms a bi-Galois extension over the quantum automorphism groups of the underlying hypergraphs. This allows us to deduce that the algebraic notion of a quantum isomorphism of hypergraphs coincides with the operational one coming from the existence of perfect quantum strategies. Viewing games themselves as hypergraphs, we analyze isomorphisms and the transfer of strategies within this setting. Finally, we construct a *-algebra whose representation theory characterizes distinct classes of quantum isomorphisms between non-local games.