Pretty good state transfer on double stars

Let A be the adjacency matrix of a graph X and suppose U(t)=exp(itA). We view A as acting on CxV(X) and take the standard basis of this space to be the vectors eu for u in V(X). hysicists say that we have perfect state transfer from vertex u to v at time τ if there is a scalar γ such that U(τ)eu=γev.(Since U(t) is unitary, ‖γ=1‖.) For example, if X is the d-cube and u and v are at distance d then we have perfect state transfer from u to v at time π/2. Despite the existence of this nice family, it has become clear that perfect state transfer is rare. Hence we consider a relaxation: we say that we have pretty good state transfer from u to v if there is a complex number γ and, for each positive real ϵ there is a time t such that ‖U(t)eu-γev‖<ϵ.Again we necessarily have |γ|=1.In a recent paper Godsil, Kirkland, Severini and Smith showed that we have have pretty good state transfer between the end vertices of the path Pn if and only n+1 is a power of two, a prime, or twice a prime. (There is perfect state transfer between the end vertices only for P2 and P3.) It is something of a surprise that the occurrence of pretty good state transfer is characterized by a number-theoretic condition. In this paper we extend the theory of pretty good state transfer. We provide what is only the second family of graphs where pretty good state transfer occurs. The graphs we use are the double-star graphs Sk,ℓ, these are trees with a vertex of degree k+1 adjacent to a vertex of degree ℓ+1, and all other vertices of degree one. We prove that perfect state transfer does not occur in any graph in this family. We show that if ℓ>2, then there is pretty good state transfer in S2,ℓ between the two end vertices adjacent to the vertex of degree three. If k,ℓ>2, we prove that there is never perfect state transfer between the two vertices of degree at least three, and we show that there is pretty good state transfer between them if and only these vertices both have degree k+1 and 4k+1 is not a perfect square. Thus we find again the the existence of perfect state transfer depends on a number theoretic condition. It is also interesting that although no double stars have perfect state transfer, there are some that admit pretty good state transfer.