Atlantic Graph Theory Seminar

Speaker: Ada Chan, York University

Title: Polygamy in state transfer

Abstract:
Let $X$ be a graph and $H$ be a Hermitian matrix associated with $X$.   The continuous-time quantum walk with Hamiltonian $H$ is
defined by the time-dependent unitary matrix
\begin{equation*}
U(t)=e^{i t H}.
\end{equation*}
Perfect state transfer occurs from vertex $a$ to vertex $b$ at time $\tau$ is $\vert U(\tau)_{b,a}\vert = 1$.   This phenomenon is relevant for information transmission in a quantum spin network.   For real and symmetric Hamiltonians, it is known that perfect state transfer can occur from a vertex to at most one other vertex,mand that graphs with perfect state transfer are rare.    

A relaxation, called pretty good state transfer, occurs from $a$ to $b$ if $\vert U(\tau)_{b,a}\vert$ gets arbitrarily close one.  Pal and Bhattacharjya discover a graph with four vertices admitting pairwise pretty good state transfer. In this talk, we present a family of graphs that admit pairwise pretty good state transfer in an arbitrarily large set of vertices. We compare this polygamous behaviour to walks with Hamiltonians that contain non-real entries.