Perfect state transfer between real pure states

TL;DR

This paper develops a comprehensive theory of perfect state transfer (PST) between real pure states in quantum spin networks, characterizing when PST occurs based on graph spectral properties and providing explicit constructions and bounds.

Contribution

It introduces new spectral characterizations of PST between real pure states and fully classifies PST in common graph families like paths and complete bipartite graphs.

Findings

Every periodic real pure state admits PST with another real pure state.

All connected graphs admit PST between some real pure states.

The minimum PST time is minimized by join graphs and specific graph constructions.

Abstract

Pure states correspond to one-dimensional subspaces of $\mathbb{C}^n$ represented by unit vectors. In this paper, we develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of a graph and prove three fundamental results: (i) every periodic real pure state $\mathbf{x}$ admits perfect state transfer with another real pure state $\mathbf{y}$, (ii) every connected graph admits perfect state transfer between real pure states, and (iii) for any pair of real pure states $\mathbf{x}$ and $\mathbf{y}$ and for any time $\tau$, there exists a real symmetric matrix $M$ such that $\mathbf{x}$ and $\mathbf{y}$ admits perfect state transfer relative to $M$ at time $\tau$. We also determine all real pure states that admit PST in complete graphs, complete bipartite graphs, paths, and cycles. This leads to a complete characterization of pair and plus state transfer in paths and complete bipartite graphs. We give constructions of graphs that admit PST between real pure states. Finally, using results on the spread of graphs, we prove that amongst all $n$-vertex simple unweighted graphs, the least minimum PST time between real pure states relative to the Laplacian is attained by any join graph, while the it is attained by the join of an empty graph and a complete graph of appropriate sizes relative to the adjacency matrix.