Algebraic Graph Theory - Chris Godsil

Monday, February 5, 2024 11:30 am - 12:30 pm EST (GMT -05:00)

Title: Laplacian State Transfer

Speaker: Chris Godsil
Affiliation: University of Waterloo
Location: Please contact Sabrina Lato for Zoom link.

Abstract: Let $X$ be a graph and let $E_1,\ldots,E_d$ be the spectral idempotents of its adjacency matrix. If $a$ and $b$ are vertices in $X$, they are \textsl{strongly cospectral} if $E_re_ae_a^TE_r = E_re_be_b^T$ for each $r$. This is a relation between the two density matrices $e_aa_a^T$ and $e_be_b^T$, and is a necessary condition for state transfer between pure states.

If $L$ is the Laplacian of a graph $X$ with $m$ edges, the matrix $(1/2m)L$ is positive semidefinite with trace one, thus it is a density matrix. We call it a \textsl{Laplacian state}. It is pure only if $X$ is an edge. We have been investigating transfer between Laplacian states in continuous quantum walks. We have extended the definition of strongly cospectral to this case, have obtained a number of results are about various forms of state transfer. My talk will be a report on this.

(This is joint work with Ada Chan, Qiuting Chen, Wanting Sun and Xiaohong Zhang.)