State Transfer and Readout Times for Trees of Diameter 4

TL;DR

This paper investigates quantum state transfer on diameter 4 trees, characterizing strongly cospectral vertices, constructing families with high-fidelity transfer, and analyzing convergence properties of transfer fidelity.

Contribution

It classifies strongly cospectral vertex pairs in diameter 4 trees and constructs explicit families with high-fidelity quantum state transfer.

Findings

Identified three types of strongly cospectral pairs in diameter 4 trees.

Constructed infinite families with pretty good state transfer.

Proved convergence properties of transfer fidelity and its derivatives.

Abstract

We consider the state transfer properties of continuous time quantum walks on trees of diameter 4. We characterize all pairs of strongly cospectral vertices in trees of diameter 4, finding that they fall into pairs of three different types. For each type, we construct an infinite family of diameter 4 trees for which there is pretty good state transfer between the pair of strongly cospectral vertices. Moreover, for two of those types, for each tree in the infinite family, we give an explicit sequence of readout times at which the fidelity of state transfer converges to $1$. For strongly cospectral vertices of the remaining type, we identify a sequence of trees and explicit readout times so that the fidelity of state transfer between the strongly cospectral vertices approaches $1.$ We also prove a result of independent interest: for a graph with the property that the fidelity of state transfer between a pair of vertices at time $t_k$ converges to $1$ as $k \rightarrow \infty,$ then the derivative of the fidelity at $t_k$ converges to $0$ as $k \rightarrow \infty. $