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

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.