State Transfer and Readout Times for Trees of Diameter 4

TL;DR

This work addresses state transfer in continuous-time quantum walks on trees with diameter 4 by classifying strongly cospectral vertex pairs into three types and constructing infinite families of diameter-4 trees that exhibit pretty good state transfer (PGST) between such pairs. The authors develop a spectral framework to characterize PST/PGST, and they derive explicit readout-time sequences that yield fidelity approaching 1 for certain configurations, providing practical timing information for quantum information transfer on tree-structured networks. A key theoretical contribution is a derivative-sensitivity result: if a fidelity sequence converges to 1, its time-derivative along that sequence must converge to 0, enhancing robustness against readout-time perturbations. Overall, the paper extends PGST results to a richer class of trees beyond paths and small trees, offering concrete constructions and readout schedules with potential applications in quantum communication topologies.

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. $

Theorems & Definitions (31)

• Lemma 1
• Theorem 2
• proof
• Example 3
• Corollary 4
• proof
• Theorem 5
• proof
• Lemma 6
• proof