When can perfect state transfer occur?

TL;DR

This paper characterizes when perfect quantum state transfer occurs in graphs, showing it is rare and linked to specific spectral properties, with implications for graph symmetry and finiteness of such graphs.

Contribution

It provides a spectral characterization of perfect state transfer, proves finiteness of graphs with this property under certain conditions, and explores symmetry implications.

Findings

Perfect state transfer implies the spectral radius is an integer or quadratic irrational.

Only finitely many graphs with maximum valency at most 4 exhibit perfect state transfer.

Graphs with perfect state transfer have cospectral vertex-deleted subgraphs and fixed points under automorphisms.

Abstract

Let $X$ be a graph on $n$ vertices with with adjacency matrix $A$ and let $H(t)$ denote the matrix-valued function $\exp(iAt)$. If $u$ and $v$ are distinct vertices in $X$, we say perfect state transfer from $u$ to $v$ occurs if there is a time $\tau$ such that $|H({\tau})_{u,v}| = 1$. Our chief problem is to characterize the cases where perfect state transfer occurs. We show that if perfect state transfer does occur in a graph, then the spectral radius is an integer or a quadratic irrational; using this we prove that there are only finitely many graphs with perfect state transfer and with maximum valency at most 4K4. We also show that if perfect state transfer from $u$ to $v$ occurs, then the graphs $X\setminus u$ and $X\setminus v$ are cospectral and any automorphism of $X$ that fixes $u$ must fix $v$ (and conversely).