On quantum perfect state transfer in weighted join graphs

TL;DR

This paper investigates conditions for perfect quantum state transfer in weighted graphs, especially join and Cartesian product graphs, revealing new cases where perfect transfer occurs or is inhibited, with implications for quantum network design.

Contribution

It extends understanding of perfect state transfer in weighted graphs, showing new classes of graphs with transfer and clarifying the impact of weights and graph operations.

Findings

Join of weighted two-vertex and regular graphs has perfect state transfer.

Half-join of weighted two-vertex and weighted regular graphs does not have perfect state transfer.

Hamming graphs exhibit perfect state transfer between all vertex pairs.

Abstract

We study perfect state transfer on quantum networks represented by weighted graphs. Our focus is on graphs constructed from the join and related graph operators. Some specific results we prove include: (1) The join of a weighted two-vertex graph with any regular graph has perfect state transfer. This generalizes a result of Casaccino et al. [clms09] where the regular graph is a complete graph or a complete graph with a missing link. In contrast, the half-join of a weighted two-vertex graph with any weighted regular graph has no perfect state transfer. This implies that adding weights in a complete bipartite graph do not help in achieving perfect state transfer. (2) A Hamming graph has perfect state transfer between each pair of its vertices. This is obtained using a closure property on weighted Cartesian products of perfect state transfer graphs. Moreover, on the hypercube, we show that perfect state transfer occurs between uniform superpositions on pairs of arbitrary subcubes. This generalizes results of Bernasconi et al. [bgs08] and Moore and Russell [mr02]. Our techniques rely heavily on the spectral properties of graphs built using the join and Cartesian product operators.