Quantum fractional revival governed by adjacency matrix Hamiltonian in unitary Cayley graphs

TL;DR

This paper characterizes when quantum fractional revival occurs in unitary Cayley graphs, linking graph structure, number theory, and quantum information transfer, and finds it only occurs when the graph has an even number of vertices.

Contribution

It provides the first characterization of quantum fractional revival in unitary Cayley graphs using spectral and number-theoretic methods, revealing conditions based on the graph's size.

Findings

Quantum fractional revival exists only for graphs with an even number of vertices.

Spectral and number-theoretic criteria for fractional revival are established.

Unitary Cayley graphs are shown to be integral and circulant, facilitating analysis.

Abstract

In this article, we give characterization for existence of quantum fractional revival in unitary Cayley graph utilizing adjacency matrix Hamiltonian. Unitary Cayley graph $X=( Z_n, S)$ is a special graph as connection set $S \subseteq Z_n$ is the collection of coprimes to $n$. Unitary Cayley graph is an integral graph and its adjacency matrix is a circulant one. We prove that quantum fractional revival in unitary Cayley graphs exists only when the number of vertices is even. Number-theoretic and spectral characterizations are given for unitary Cayley graph admitting quantum fractional revival. Quantum fractional revival is analogous to quantum entanglement. It is one of qubit state transfer phenomena useful in communication of quantum information.