Quantum state transfer between twins in weighted graphs

TL;DR

This paper investigates how twin vertices in weighted graphs influence quantum state transfer, providing characterizations for various transfer types and applying these to specific graph structures.

Contribution

It offers new characterizations of quantum state transfer phenomena between twin vertices in weighted graphs, extending understanding beyond unweighted cases.

Findings

Characterizations of periodicity, perfect state transfer, and pretty good state transfer between twin vertices.

Application of these characterizations to double cones on regular graphs.

Insights into the role of graph weights and structures in quantum information transfer.

Abstract

Twin vertices in simple unweighted graphs are vertices that have the same neighbours and, in the case of weighted graphs with possible loops, the corresponding incident edges have equal weights. In this paper, we explore the role of twin vertices in quantum state transfer. In particular, we provide characterizations of periodicity, perfect state transfer, and pretty good state transfer between twin vertices in a weighted graph with respect to its adjacency, Laplacian and signless Laplacian matrices. As an application, we provide characterizations of all simple unweighted double cones on regular graphs that exhibit periodicity, perfect state transfer, and pretty good state transfer.