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Fractional revival on quasi-abelian Cayley graphs

Yi Fang, Xueyi Huang, Xiaogang Liu, Xiongfeng Zhan

TL;DR

The paper addresses fractional revival in quasi-abelian Cayley graphs, extending results beyond abelian groups. It derives a necessary and sufficient condition for revival via the irreducible characters of $G$, showing that revival requires a central involution and specific eigenphase alignments, which in particular implies that all eigenvalues $\lambda_\chi$ are integers. As a consequence, Cao–Luo’s abelian case is recovered and nonexistence results are obtained for odd-order groups and for $G = S_n$ with $n \ge 3$; the paper also provides an explicit example on $G = \mathbb{Z}_6 \times D_3$ that exhibits revival.

Abstract

Fractional revival, a quantum transport phenomenon critical to entanglement generation in quantum spin networks, generalizes the notion of perfect state transfer on graphs. A Cayley graph $\mathrm{Cay}(G,S)$ is called quasi-abelian if its connection set $S$ is a union of conjugacy classes of the group $G$. In this paper, we establish a necessary and sufficient condition for quasi-abelian Cayley graphs to have fractional revival. This extends a result of Cao and Luo (2022) on the existence of fractional revival in Cayley graphs over abelian groups.

Fractional revival on quasi-abelian Cayley graphs

TL;DR

The paper addresses fractional revival in quasi-abelian Cayley graphs, extending results beyond abelian groups. It derives a necessary and sufficient condition for revival via the irreducible characters of , showing that revival requires a central involution and specific eigenphase alignments, which in particular implies that all eigenvalues are integers. As a consequence, Cao–Luo’s abelian case is recovered and nonexistence results are obtained for odd-order groups and for with ; the paper also provides an explicit example on that exhibits revival.

Abstract

Fractional revival, a quantum transport phenomenon critical to entanglement generation in quantum spin networks, generalizes the notion of perfect state transfer on graphs. A Cayley graph is called quasi-abelian if its connection set is a union of conjugacy classes of the group . In this paper, we establish a necessary and sufficient condition for quasi-abelian Cayley graphs to have fractional revival. This extends a result of Cao and Luo (2022) on the existence of fractional revival in Cayley graphs over abelian groups.

Paper Structure

This paper contains 3 sections, 6 theorems, 50 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Lemma 2.1

(See also CL22) Let $G$ be a finite group, and let $\Gamma$ be a connected Cayley graph over $G$. Suppose that $\alpha$ and $\beta$ are two complex numbers satisfying $\beta\neq 0$ and $|\alpha|^2+|\beta|^2=1$. Then $(\alpha,\beta)$-revival occurs on $\Gamma$ from a vertex $u$ to a vertex $v$ at tim where $Q$ is a symmetric permutation matrix with zero diagonals and $\alpha\overline{\beta} + \over

Theorems & Definitions (12)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • ...and 2 more