Quantum Fractional Revival and Entanglement Entropy in Unitary Cayley Graphs

Abstract

This paper extends the theory of quantum fractional revival (QFR) on unitary Cayley graphs $X=(V(\mathbb{Z}_n),E(S))$ in several directions that remained unresolved in previous work. First, we investigate QFR with respect to the Laplacian matrix Hamiltonian in addition to the adjacency matrix Hamiltonian. In particular, we prove that for regular graphs the two models differ only by a global phase factor, and we determine the conditions under which the Laplacian framework independently admits QFR. Second, for unitary Cayley graphs of order $n=2p$, where $p$ is an odd prime, we derive an explicit closed-form expression for the minimum revival time, $t^{*}=\frac{2π}{p},$ and show that the associated revival amplitudes are given by \[ α=\cos\!\left(\frac{2π}{p}\right), \qquad β=-i\sin\!\left(\frac{2π}{p}\right). \] Third, we provide a complete characterization of strongly cospectral vertex pairs in $X=(V(\mathbb{Z}_n),E(S))$ through the arithmetic structure of $\mathbb{Z}_n$, establishing that strong cospectrality is equivalent to antipodality whenever $n$ is twice a prime. Finally, we compute the von Neumann entanglement entropy generated by QFR for all admissible graphs, thereby obtaining a collection of quantum information measures and proving that the entropy depends solely on the revival amplitudes $|α|$ and $|β|$.