Perfect State Transfer on Signed Graphs

TL;DR

This paper demonstrates that signing edges in graphs, especially negative edges, can enable perfect quantum state transfer in cases where unsigned graphs cannot, introducing new constructions and insights for quantum information transfer.

Contribution

It shows how negative edges in signed graphs can facilitate perfect state transfer, expanding possibilities beyond unsigned graphs and introducing new graph constructions.

Findings

Signed join of negative 2-clique with positive regular graph achieves perfect state transfer.

Signed complete graphs can have perfect state transfer if conditions on subgraphs are met.

Double-cover of signed graphs can exhibit perfect state transfer under certain conditions.

Abstract

We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. Specific results we prove include: (1) The signed join of a negative 2-clique with any positive (n,3)-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as n increases. (2) A signed complete graph has perfect state transfer if its positive subgraph is a regular graph with perfect state transfer and its negative subgraph is periodic. This shows that signing is useful for creating perfect state transfer since no complete graph (except for the 2-clique) has perfect state transfer. (3) The double-cover of a signed graph has perfect state transfer if the positive subgraph has perfect state transfer and the negative subgraph is periodic. Here, signing is useful for constructing unsigned graphs with perfect state transfer. Furthermore, we study perfect state transfer on a family of signed graphs called the exterior powers which is derived from a many-fermion quantum walk on graphs.