Perfect state transfer on graphs with a potential

TL;DR

This paper investigates quantum state transfer on graphs with potentials, showing limitations on paths and conditions for transfer with potentials, including new examples and graph product analyses.

Contribution

It demonstrates that potentials enable perfect state transfer in cases where it is otherwise impossible, such as certain graphs with shared neighborhoods.

Findings

No perfect state transfer on paths longer than three without potential

Potential enables perfect state transfer between vertices with common neighborhoods

Graph products can facilitate perfect state transfer with potentials

Abstract

In this paper we study quantum state transfer (also called quantum tunneling) on graphs when there is a potential function on the vertex set. We present two main results. First, we show that for paths of length greater than three, there is no potential on the vertices of the path for which perfect state transfer between the endpoints can occur. In particular, this answers a question raised by Godsil in Section 20 of [8]. Second, we show that if a graph has two vertices that share a common neighborhood, then there is a potential on the vertex set for which perfect state transfer will occur between those two vertices. This gives numerous examples where perfect state transfer does not occur without the potential, but adding a potential makes perfect state transfer possible. In addition, we investigate perfect state transfer on graph products, which gives further examples where perfect state transfer can occur.