Pretty good fractional revival on abelian Cayley graphs
Akash Kalita, Bikash Bhattacharjya
TL;DR
The paper develops a comprehensive spectral framework for PGFR on abelian Cayley graphs, giving a necessary-and-sufficient condition in terms of parity (requiring $n$ even and $b-a$ of order two) plus Kronecker-approximation-type constraints on eigenvalues. It translates these conditions into circulant and non-circulant cases, deriving new PGFR results for families on $n=2p^s$ and related constructions, as well as results about complements and nonexistence in several regimes. By constructing infinite families of graphs that exhibit PGFR without FR or PGST, the authors extend and generalize prior work on cycles and paths (Chan et al.) to a broad class of abelian Cayley graphs. The work advances understanding of fractional revival phenomena in quantum walks on structured graphs and provides tools for designing graphs with targeted PGFR properties. The results have potential implications for entanglement generation and quantum routing in networked quantum systems modeled by abelian Cayley graphs.
Abstract
Let $Γ$ be a graph with the adjacency matrix $A$. The transition matrix of $Γ$, denoted $H(t)$, is defined as $H(t) := \exp(-\textbf{i}tA)$, where $\textbf{i} := \sqrt{-1}$ and $t$ is a real variable. The graph $Γ$ is said to exhibit fractional revival (FR in short) between the vertices $a$ and $b$ if there exists a positive real number $t$ such that $H(t){\textbf{e}_{a}} = α{\textbf{e}_{a}} + β{\textbf{e}_{b}}$, where $α, β\in \mathbb{C}$ such that $β\neq 0$ and $|α|^2 + |β|^2 = 1$. The graph $Γ$ is said to exhibit pretty good fractional revival (PGFR in short) between the vertices $a$ and $b$ if there exists a sequence of real numbers $\{t_k\}$ with $\lim_{k\to\infty} H(t_k){\textbf{e}_{a}} = α{\textbf{e}_{a}} + β{\textbf{e}_{b}}$, where $α, β\in \mathbb{C}$ such that $β\neq 0$ and $|α|^2 + |β|^2 = 1$. In the definition of PGFR, if $α=0$ then $Γ$ is said to exhibit pretty good state transfer (PGST in short) between $a$ and $b$. In this paper, we obtain some sufficient conditions for circulant graphs exhibiting PGFR. We also find some sufficient conditions for non-circulant abelian Cayley graphs exhibiting PGFR. From these sufficient conditions, we find infinite families of circulant graphs and non-circulant abelian Cayley graphs exhibiting PGFR that fail to exhibit FR and PGST. Finally, we obtain some necessary conditions for some families of circulant graphs exhibiting PGFR. Some of our results generalize the results of Chan et al. [Pretty good quantum fractional revival in paths and cycles. \textit {Algebr. Comb.} 4(6) (2021), 989-1004.] for cycles.