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The hypergraph isomorphism game, Hopf algebras and Galois extensions

Georgios Baziotis, Alexandros Chatzinikolaou, Gage Hoefer

TL;DR

The paper develops an algebraic-operational framework for quantum isomorphisms of hypergraphs using compact quantum groups and Hopf $*$-algebras, introducing a synchronous hypergraph isomorphism game whose $*$-algebra encodes all major isomorphism notions. It proves that hypergraph quantum isomorphisms can exist without classical isomorphisms, and that the HyperIso game algebra is a quotient of the graph isomorphism game algebra, linking hypergraph and graph quantum symmetries. A bi-Galois extension structure is established for the hypergraph isomorphism space, connecting algebraic and operational perspectives and enabling transfer of strategies between games via invariant states. The work also develops a universal no-signalling framework for non-local games, showing that NS algebras provide a robust setting for characterizing and transporting quantum strategies across games, thereby unifying several notions of quantum isomorphism under a common bi-Galois-Hopf-algebraic umbrella.

Abstract

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.

The hypergraph isomorphism game, Hopf algebras and Galois extensions

TL;DR

The paper develops an algebraic-operational framework for quantum isomorphisms of hypergraphs using compact quantum groups and Hopf -algebras, introducing a synchronous hypergraph isomorphism game whose -algebra encodes all major isomorphism notions. It proves that hypergraph quantum isomorphisms can exist without classical isomorphisms, and that the HyperIso game algebra is a quotient of the graph isomorphism game algebra, linking hypergraph and graph quantum symmetries. A bi-Galois extension structure is established for the hypergraph isomorphism space, connecting algebraic and operational perspectives and enabling transfer of strategies between games via invariant states. The work also develops a universal no-signalling framework for non-local games, showing that NS algebras provide a robust setting for characterizing and transporting quantum strategies across games, thereby unifying several notions of quantum isomorphism under a common bi-Galois-Hopf-algebraic umbrella.

Abstract

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.
Paper Structure (14 sections, 24 theorems, 180 equations, 1 figure)

This paper contains 14 sections, 24 theorems, 180 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Non-local games and correlation sets
  4. Compact quantum groups and Hopf algebras
  5. Quantum automorphisms of hypergraphs
  6. The hypergraph isomorphism game
  7. Game description
  8. Presentation of the game algebra
  9. Connection to the graph isomorphism game
  10. Bi-Galois extensions
  11. The game isomorphism game and transfer of strategies
  12. The NS game $*$-algebra
  13. Simulation paradigm
  14. The NS algebra as a quotient bi-Galois extension

Key Result

Theorem 2.1

Let $\mathcal{G} = (X, A, \lambda)$ be a synchronous game. Then

Figures (1)

  • Figure 1: Implication relations between isomorphisms of $\Lambda_1, \Lambda_2$.

Theorems & Definitions (55)

  • Theorem 2.1: hmpskps
  • Remark 2.2
  • Remark 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof : (i) $\Leftrightarrow$ (ii)
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Theorem 3.6
  • ...and 45 more