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Quantum isomorphism of $2$-graphs

Soumalya Joardar, Atibur Rahaman, Jitender Sharma

TL;DR

This work defines and analyzes the quantum automorphism group of rank-$2$ graphs by introducing defining triples $(\mathcal{G}_{1},\mathcal{G}_{2},\theta)$ and constructing the corresponding quantum automorphism group ${\rm Aut}^{+}(\mathcal{G}_{1},\mathcal{G}_{2},\theta)$. It then defines quantum isomorphism between pairs of $2$-graphs via quantum equivalence of their defining triples and proves that equivalent triples yield isomorphic quantum automorphism groups; a practical criterion ties quantum isomorphism to the automorphism group of the disjoint union. The authors provide a non-trivial example of two $2$-graphs that are not quantum isomorphic, and extend known results on free wreath products to the $2$-graph setting. The work also establishes a structural decomposition for automorphism groups of disjoint unions in terms of free products and free wreath products, enriching the interplay between higher-rank graphs and compact quantum groups with potential connections to nonlocal games and quantum information.

Abstract

We formulate a notion of the quantum automorphism group of a $2$-graph. After some preliminary computations, we define quantum isomorphism between a pair of $2$-graphs. We produce a `non-trivial' example of a pair of $2$-graphs that are not quantum isomorphic to each other.

Quantum isomorphism of $2$-graphs

TL;DR

This work defines and analyzes the quantum automorphism group of rank- graphs by introducing defining triples and constructing the corresponding quantum automorphism group . It then defines quantum isomorphism between pairs of -graphs via quantum equivalence of their defining triples and proves that equivalent triples yield isomorphic quantum automorphism groups; a practical criterion ties quantum isomorphism to the automorphism group of the disjoint union. The authors provide a non-trivial example of two -graphs that are not quantum isomorphic, and extend known results on free wreath products to the -graph setting. The work also establishes a structural decomposition for automorphism groups of disjoint unions in terms of free products and free wreath products, enriching the interplay between higher-rank graphs and compact quantum groups with potential connections to nonlocal games and quantum information.

Abstract

We formulate a notion of the quantum automorphism group of a -graph. After some preliminary computations, we define quantum isomorphism between a pair of -graphs. We produce a `non-trivial' example of a pair of -graphs that are not quantum isomorphic to each other.
Paper Structure (8 sections, 12 theorems, 76 equations, 1 figure)

This paper contains 8 sections, 12 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Higher rank graphs
  4. Compact quantum groups and their actions
  5. Quantum automorphism group of finite graphs
  6. Main Results
  7. Examples and Computations I
  8. Examples and computations II: Free product and free wreath product

Key Result

Lemma 2.18

The underlying $C^{\ast}$-algebra of the quantum automorphism group ${\rm Aut}^{+}(\mathcal{G})$ of a finite graph $\mathcal{G}$ with $m$ edges and $n$ vertices (without multiple edges) is the universal $C^{\ast}$-algebra generated by $\{q_{ij}\}_{i,j=1,\ldots,n}$ satisfying the following relations: The coproduct on the generators is given by $\Delta(q_{ij})=\sum_{k=1}^{n}q_{ik}\otimes q_{kj}$. Th

Figures (1)

  • Figure :

Theorems & Definitions (45)

  • Definition 2.2: see NYJM*Definition 1.1
  • Remark 2.3
  • Definition 2.4: see NYJM*Definition 1.9
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.10
  • Remark 2.11
  • Definition 2.12
  • ...and 35 more