Pretty Good State Transfer on Circulant Graphs

TL;DR

This paper characterizes when pretty good state transfer occurs in circulant graphs, especially cycles, and identifies new classes of graphs, including certain unions and complements, that admit this quantum information transfer property.

Contribution

It provides a complete characterization of pretty good state transfer in cycles and extends results to unions and complements of circulant graphs, including non-circulant cases.

Findings

Pretty good state transfer occurs in cycles if and only if the number of vertices is a power of two.

Unions of integral circulant graphs with cycles on 2^k vertices admit pretty good state transfer.

Some complements of cycles also admit pretty good state transfer, while others do not.

Abstract

Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H(t):=\exp{\left(-itA\right)},\;t\in\Rl$. The graph $G$ is said to admit pretty good state transfer between a pair of vertices $u$ and $v$ if there exists a sequence of real numbers $\{t_k\}$ and a complex number $\gamma$ of unit modulus such that $\lim\limits_{k\rightarrow\infty} H(t_k) e_u=\gamma e_v.$ We find that pretty good state transfer occurs in a cycle on $n$ vertices if and only if $n$ is a power of two and it occurs between every pair of antipodal vertices. In addition, we look for pretty good state transfer in more general circulant graphs. We prove that union (edge disjoint) of an integral circulant graph with a cycle, each on $2^k$ $(k\geq 3)$ vertices, admits pretty good state transfer. The complement of such union also admits pretty good state transfer. This enables us to find some non-circulant graphs admitting pretty good state transfer. Among the complement of cycles we also find a class of graphs not exhibiting pretty good state transfer.