Quantum fractional revival on zero-divisor graphs over $\mathbb{Z}_n$

TL;DR

This paper investigates quantum state transfer phenomena on zero-divisor graphs over Z_n, providing conditions for perfect state transfer and fractional revival based on graph partitions and spectral properties.

Contribution

It offers a new characterization of fractional revival and perfect state transfer on zero-divisor graphs using equitable partitions and spectral analysis.

Findings

Derived a sufficient condition on n for perfect state transfer.

Showed fractional revival occurs only in cells of size 2 within the partition.

Proved fractional revival does not occur on Z_{p^2q}.

Abstract

In this paper, we characterize the existence of perfect state transfer (PST) and fractional revival in continuous-time quantum walks on the zero-divisor graph $\Gamma(\mathbb{Z}_n)$. By using the canonical equitable partition of $\Gamma(\mathbb{Z}_n)$ induced by the proper divisors of $n$, we derive a sufficient condition on $n$ for PST to occur between a pair of vertices. We show that fractional revival is restricted to cells of size $2$ within the equitable partition. Furthermore, assuming $-1$ is not an eigenvalue of the quotient spectrum, we establish that two vertices in $\Gamma(\mathbb{Z}_n)$ are strongly cospectral if and only if they form a cell of size $2$ within the equitable partition that is either a set of false twins or true twins. Finally, we provide a characterization of fractional revival on bipartite $\Gamma(\mathbb{Z}_n)$ and prove the non-existence of fractional revival on $\Gamma(\mathbb{Z}_{p^2q})$.