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 ellG, packaged as the 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 \[ tn\!\leftG, tfrac{1}{m}right = \frac{1}{m^{n-1}}\, tnG, \] valid whenever the initial edge state is uniform commutative, where n=|VG|, m=|EG|, and tn\(G, w\) denotes the weighted spanning-tree count. We further identify a structural mechanism - the -2 eigenspace of ellG - 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.