Schur States, Average Mixing, and Counting Trees on Line Graphs' CTQW

TL;DR

This paper introduces a family of complex weights from quantum walks on line graphs, relating weighted spanning-tree counts to original graphs and identifying structural states with entropy preservation.

Contribution

It defines the Schur state for quantum walks, establishes a formula linking weighted spanning-tree counts, and characterizes states with entropy invariance beyond regular graphs.

Findings

Weighted spanning-tree count scales with initial state and graph parameters.

Structural states are linked to the eigenspace of the line graph, especially for Eulerian graphs.

Commutative states are characterized by von Neumann entropy preservation.

Abstract

We introduce a family of complex-valued edge weights on a finite simple graph $\G$ arising from a continuous-time quantum walk on the line graph $\ell\G$, packaged as the \emph{Schur state}: an $n \times n$ Hermitian matrix encoding the amplitudes of an edge-state walk. The entrywise modulus square induces a real-weighted adjacency matrix $A(e)$ and Laplacian $L(e)$, and time-averaging yields a weighted graph whose spanning-tree count we relate to that of $\G$. Our main result is \[ tn\!\left(\G, \tfrac{1}{m}\right) = \frac{1}{m^{n-1}}\, tn(\G), \] valid whenever the initial edge state is \emph{uniform commutative}, where $n=|V\G|$, $m=|E\G|$, and $tn(\G, w)$ denotes the weighted spanning-tree count. We further identify a structural mechanism -- the $-2$ eigenspace of $\ell\G$ -- providing uniform commutative states beyond the regular case, in particular for line graphs of Eulerian graphs with an even number of edges. As a side result, we establish that commutative states are precisely the states whose von Neumann entropy is preserved under average mixing.