Perfect state transfer from vertex 
to a distinct vertex 
of a graph
occurs at time 
when
 |
where 
is the adjacency matrix for an undirected
graph, or the Hermitian adjacency matrix
for a mixed graph. Equivalently, the evolution maps
the coordinate vector at 
to a scalar of absolute value
1 times the coordinate vector at 
. This is a spectral condition on the matrix
exponential (Song and Lin 2026).
For the two-vertex complete graph, 
, and
 |
Thus perfect state transfer occurs at 
. In an undirected graph,
the transition matrix is symmetric, so transfer from 
to 
implies transfer from 
to 
at the same time. For a mixed graph
this same-time symmetry can fail (Song and Lin 2026).
