Real State Transfer

TL;DR

This paper investigates quantum walks on graphs, revealing that perfect state transfer occurs when the density matrix is real and uniform mixing when entries are algebraic, leading to restrictions on certain graph classes.

Contribution

It establishes conditions under which perfect state transfer and uniform mixing occur, linking these phenomena to properties of the density matrix and graph structure.

Findings

Perfect state transfer occurs when the density matrix is real.

Uniform mixing is associated with algebraic entries in the density matrix.

Restrictions are derived for uniform mixing on bipartite and oriented graphs.

Abstract

A continuous quantum walk on a graph $X$ with adjacency matrix $A$ is specified by the 1-parameter family of unitary matrices $U(t)=\exp(itA)$. These matrices act on the state space of a quantum system, the states of which we may represent by density matrices, positive semidefinite matrices with rows and columns indexed by $V(X)$ and with trace $1$. The square of the absolute values of the entries of a column of $U(t)$ define a probability density on $V(X)$, and it is precisely these densities that predict the outcomes of measurements. There are two special cases of physical interest: when the column density is supported on a vertex, and when it is uniform. In the first case we have perfect state transfer; in the second, uniform mixing. There are many results concerning state transfer and uniform mixing. In this paper we show that these results on perfect state transfer hold largely because at the time it occurs, the density matrix is real. We also show that the results on uniform mixing obtained so far hold because the entries of the density matrix are algebraic numbers. As a consequence of these we derive strong restrictions on the occurence of uniform mixing on bipartite graphs and on oriented graphs.