Pretty good state transfer in graphs with an involution

TL;DR

This paper demonstrates that graphs with an involution can be engineered with specific potentials to achieve pretty good quantum state transfer between symmetric vertices, including paths of arbitrary length, contrasting with limitations on perfect transfer.

Contribution

It introduces a method to induce pretty good quantum state transfer in graphs with involution by selecting suitable potentials, even in cases where perfect transfer is impossible.

Findings

Pretty good state transfer can be achieved in graphs with involution.

Potential can be non-zero only at the vertices involved in transfer.

Paths of any length can exhibit pretty good state transfer with appropriate potentials.

Abstract

We study pretty good quantum state transfer (i.e., state transfer that becomes arbitrarily close to perfect) between vertices of graphs with an involution in the presence of an energy potential. In particular, we show that if a graph has an involution that fixes at least one vertex or at least one edge, then there exists a choice of potential on the vertex set of the graph for which we get pretty good state transfer between symmetric vertices of the graph. We show further that in many cases, the potential can be chosen so that it is only non-zero at the vertices between which we want pretty good state transfer. As a special case of this, we show that such a potential can be chosen on the endpoints of a path to induce pretty good state transfer in paths of any length. This is in contrast to the result of [6], in which the authors show that there cannot be perfect state transfer in paths of length 4 or more, no matter what potential is chosen.