Average mixing of continuous quantum walks

If X is a graph with adjacency matrix A, then we define H ( t ) to be the operator exp ( i t A ) . The Schur (or entrywise) product H ( t ) ∘ H ( − t ) is a doubly stochastic matrix and because of work related to quantum computing, we are concerned with the average mixing matrix M ˆ X , defined by M ˆ X = lim T → ∞ 1 T ∫ 0 T H ( t ) ∘ H ( − t ) d t . In this paper we establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We see that in a number of cases its form is surprisingly simple. Thus for the path on n vertices it is equal to 1 2 n + 2 ( 2 J + I + T ) where T is the permutation matrix that swaps j and n + 1 − j for each j. If X is an odd cycle or, more generally, if X is one of the graphs in a pseudocyclic association scheme on n vertices with d classes, each of valency m, then its average mixing matrix is n − m + 1 n 2 J + m − 1 n I . (One reason this is interesting is that a graph in a pseudocyclic scheme may have trivial automorphism group, and then the mixing matrix is more symmetric than the graph itself.)