Quantum fractional revival on unitary Cayley graphs over finite commutative rings
Saowalak Jitngam, Poom Kumam, Songpon Sriwongsa
TL;DR
The paper addresses QFR on unitary Cayley graphs $ ext{G}_R$ over finite commutative rings with identity by leveraging the spectral decomposition of adjacency matrices and the structure of local rings. It establishes a complete classification for finite local rings: $ ext{G}_R$ exhibits QFR if and only if $R$ is $ ext{F}_2$, $ ext{Z}_4$, or $ ext{Z}_2[x]/(x^2)$, using explicit eigenvalues and projections. For rings that decompose as products of local rings, the authors analyze $ ext{G}_R$ via tensor products, deriving necessary conditions such as $|R|$ being even and each maximal ideal size $|M_j| ext{ in }\\{1,2\}$; they provide partial results relating QFR to PST and demonstrate phenomena through concrete examples like $ ext{Z}_6$. The results illuminate how ring-theoretic structure governs quantum state revival on unitary Cayley graphs and offer groundwork for further exploration of QFR in composite ring settings and related graph products. The work advances understanding of QFR in algebraic graph models with potential implications for quantum information transfer on ring-based networks.
Abstract
In this paper, we investigate the existence of quantum fractional revival in unitary Cayley graphs over finite commutative rings with identity. We characterize all finite local rings that permit quantum fractional revival in their unitary Cayley graphs. Additionally, we present results for the case of finite commutative rings, as they can be expressed as products of finite local rings.