Quantum fractional revival on graphs

Abstract

We develop a general spectral framework to analyze quantum fractional revival in quantum spin networks. In particular, we introduce generalizations of the notions of cospectral and strongly cospectral vertices to arbitrary subsets of vertices, and give various examples. This work resolves two open questions of Chan et. al. ["Quantum Fractional Revival on graphs". Discrete Applied Math

Laplacian Fractional Revival on Graphs

“…In stark contrast to perfect state transfer, which is known to be monogamous, one of our families exhibit polygamous fractional revival. This unexpected result answers an open question in [6].…”

“…We remark that, as these graphs are regular, they are also the first infinite family of unweighted graphs that admit polygamous adjacency fractional revival. This answers an open question in [6].…”

“…A lot of theory has been developed for the adjacency model; for example, there is a complete characterization of adjacency perfect state transfer in terms of the spectrum of A [13], and one can decide in polynomial time whether this phenomenon occurs on a particular graph [10]. It is also known that both perfect state transfer and fractional revival are rare with respect to the adjacency matrix [13,6].…”

Pretty good quantum fractional revival in paths and cycles

“…Most notably, the notion of pretty good state transfer was introduced by Godsil [15] and Vinet et al [26]. There is pretty good state transfer from It was shown in [6] that if there is fractional revival from u to v, then there is fractional revival from v to u at the same time. Thus, we will simply say there is fractional revival between u and v at time t. Where perfect state transfer represents the exact transfer of a quantum state between two nodes, fractional revival represents the transfer of a quantum state to a superposition over exactly two nodes.…”

“…Thus, we will simply say there is fractional revival between u and v at time t. Where perfect state transfer represents the exact transfer of a quantum state between two nodes, fractional revival represents the transfer of a quantum state to a superposition over exactly two nodes. Fractional revival is important in the study of entanglement generation [6]. Fractional revival in weighted paths is studied in [2,6,7,9,13,14].…”