PERFECT STATE TRANSFER IN UNITARY CAYLEY GRAPHS OVER LOCAL RINGS
Yotsanan Meemark, Songpon Sriwongsa, Communicated Dariush Kiani · 2014
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.