Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.01953 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910187542020096 |
|---|---|
| author | Kang, Musung |
| author_facet | Kang, Musung |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01953 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Schur States, Average Mixing, and Counting Trees on Line Graphs' CTQW Kang, Musung Quantum Physics Combinatorics 05C50, 05C30, 05C76, 81P68 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. |
| title | Schur States, Average Mixing, and Counting Trees on Line Graphs' CTQW |
| topic | Quantum Physics Combinatorics 05C50, 05C30, 05C76, 81P68 |
| url | https://arxiv.org/abs/2605.01953 |