In this article, we give characterization for existence of quantum fractional revival in unitary Cayley graph utilizing adjacency matrix Hamiltonian. Unitary Cayley graph
X=(Zn,S) is a special graph as connection set
S⊆Zn 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.
This paper extends the theory of quantum fractional revival (QFR) on unitary Cayley graphs
X=(V(Zn),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∗=p2π, and show that the associated revival amplitudes are given by
α=cos(p2π),β=−isin(p2π). Third, we provide a complete characterization of strongly cospectral vertex pairs in
X=(V(Zn),E(S)) through the arithmetic structure of
Zn, 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∣.
Fractional revival, known as a quantum transport phenomenon, is essential for entanglement generation in quantum spin networks. The concept of fractional revival is a generalization of perfect state transfer and periodicity on graphs. In this paper, we propose a sufficient and necessary condition for abelian Cayley graphs having fractional revival between any two distinct vertices. With this characterization, two general constructions of abelian Cayley graphs having fractional revival is presented. Meanwhile, we establish several new families of abelian Cayley graphs admitting fractional revival.
Fractional revival, a quantum transport phenomenon critical to entanglement generation in quantum spin networks, generalizes the notion of perfect state transfer on graphs. A Cayley graph
Cay(G,S) is called quasi-abelian if its connection set
S is a union of conjugacy classes of the group
G. In this paper, we establish a necessary and sufficient condition for quasi-abelian Cayley graphs to have fractional revival. This extends a result of Cao and Luo (2022) on the existence of fractional revival in Cayley graphs over abelian groups.
Fractional revival (FR), a generalization of perfect state transfer (PST), is a significant phenomenon in quantum state transfer that allows quantum information to be transmitted between two qubits with a certain probability. The existence of FR has been extensively studied on many classes of graphs. However, oriented graphs have not yet been investigated. In this paper, we investigate the existence of proper FR on oriented graphs. We first establish necessary and sufficient conditions for oriented graphs to admit proper FR between strongly cospectral vertices. Furthermore, we prove that oriented Cayley graphs over abelian groups do not admit proper FR, and we subsequently characterize the conditions under which oriented semi-Cayley graphs over abelian groups admit proper FR.
Abstract. In this work, using eigenvalues and eigenvectors of unitary Cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allow a PST occurring in its unitary Cayley graph. Moreover, we have some developments when R is a product of local rings. 1. Perfect State Transfer and Unitary Cayley Graphs Let G be an undirected graph whose vertex set V (G) = {v1,..., vn}. The adjacency matrix of G, written AG, is the n×n matrix in which entry ajk is the number of edges in G with endpoint {vj, vk}. Define the matrix-valued function H(t) = exp(itAG) for all t ≥ 0. We say there is a perfect state transfer (PST) from vertex vj to vertex vk if there is a time t such that |H(t)jk | = 1. We note that our matrix H(t) determines what is known in graph theory as a continuous quantum walk. For background on quantum walks, we refer the reader to [9] and [10]. A perfect state transfer in continuous-time quantum walk on graphs has received considerable attention in quantum information and computations in Physics (e.g., [2, 4]). An excellent survey of perfect state transfer graphs and related questions are given by Godsil [8]. Observe that H(t) has the following properties: (i) H(t) is symmetric, (ii) H(t) = H(t)−1, where ¯ is the complex conjugate, (iii) H(t) is unitary, i.e., (H(t))T = H(t)−1. Thus, we have the next proposition.