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On quantum symmetries of graphs

Olha Ostrovska, Vasyl Ostrovskyi, Ludmila Turowska

Abstract

Let $G$ be a simple finite graph, and let $\mathcal U_G$ be the related quantum graph. We study the game algebra $C(\mathrm{Qut}(\mathcal U_G))$ of quantum automorphism of $\mathcal U_G$. Moreover, we prove that for any graph $G$ with $|V(G)|\geq 3$, the quantum graph $\mathcal U_G$ admits nonlocal symmetry, meaning that there exists a perfect quantum no-signaling correlation

On quantum symmetries of graphs

Abstract

Let be a simple finite graph, and let be the related quantum graph. We study the game algebra of quantum automorphism of . Moreover, we prove that for any graph with , the quantum graph admits nonlocal symmetry, meaning that there exists a perfect quantum no-signaling correlation
Paper Structure (8 sections, 15 theorems, 69 equations)

This paper contains 8 sections, 15 theorems, 69 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Game $C^*$-algebra for complete graphs
  3. Structure of the game algebra $C(\mathrm{Qut}(\mathcal{U}_{K_n}))$
  4. Properties of $C(\mathrm{Qut}(\mathcal{U}_{K_2}))$ and $C(\mathrm{Qut}(\mathcal{U}_{K_3}))$
  5. Nonlocal symmetries
  6. Nonlocal symmetries for $K_3$
  7. Decomposition into regular graphs
  8. Nonlocal symmetries for higher order graphs.

Key Result

Theorem 1

For each $n\in\mathbb N$, $C(\mathrm{Qut}(\mathcal{U}_{K_n}))$ is the universal $C^*$-algebra generated by $u_{i,j}^*u_{k,l}$, $i,j,k,l\in \mathbb N_n$ where each $u_{i,j}$ is a partial isometry and $U=(u_{i,j})_{i,j=1}^n$ is a bi-unitary. The map $\varphi:C(S_n^+)\to C(\mathrm{Qut}(\mathcal{U}_{K_n

Theorems & Definitions (30)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • ...and 20 more