Pretty good state transfer between internal nodes of paths

TL;DR

This paper demonstrates that certain path graphs with specific lengths exhibit pretty good state transfer between internal nodes, expanding understanding of quantum walk behavior beyond extremal vertices.

Contribution

It provides the first examples of pretty good state transfer between internal vertices on a path, for paths of length 2^t p - 1 with odd prime p.

Findings

Pretty good state transfer occurs between internal vertices in specified path graphs.

Such transfer is demonstrated for paths of length 2^t p - 1, with p an odd prime.

This is the first known case of internal vertex transfer not involving extremal vertices.

Abstract

We study a continous-time quantum walk on a path graph. In this paper, we show that, for any odd prime $p$ and positive integer $t$, the path on $2^t p - 1$ vertices admits pretty good state transfer between vertices $a$ and $n+1-a$ for each $a$ that is a multiple of $2^{t-1}$ with respect to the quantum walk model determined by the XY-Hamiltonian. This gives the first examples of pretty good state transfer occurring between internal vertices on a path, when it does not occur between the extremal vertices.