# Pair State Transfer

## TL;DR

This paper investigates perfect pair state transfer in quantum walks on graphs, providing new constructions, characterizations, and insights into the phenomena, especially contrasting with vertex state transfer.

## Contribution

It introduces methods to construct infinite families of graphs with perfect pair transfer and characterizes this transfer on paths and cycles, highlighting differences from vertex transfer.

## Key findings

- Constructed infinite families of graphs with perfect pair transfer
- Characterized perfect pair transfer on paths and cycles
- Identified equivalence of plus and pair state transfer in bipartite graphs

## Abstract

Let $L$ denote the Laplacian matrix of a graph $G$. We study continuous quantum walks on $G$ defined by the transition matrix $U(t)=\exp\left(itL\right)$. The initial state is of the pair state form, $e_a-e_b$ with $a,b$ being any two vertices of $G$. We provide two ways to construct infinite families of graphs that have perfect pair transfer. We study a "transitivity" phenomenon which cannot occur in vertex state transfer. We characterize perfect pair state transfer on paths and cycles. We also study the case when quantum walks are generated by the unsigned Laplacians of underlying graphs and the initial states are of the plus state form, $e_a+e_b$. When the underlying graphs are bipartite, plus state transfer is equivalent to pair state transfer.