Sensitivity analysis of perfect state transfer in quantum spin networks

For a weighted graph G with adjacency matrix A  , let U(t)=eitAU(t)=eitA. For indices s,rs,r, the fidelity of transfer at time t   is p(t)=|u(t)s,r|2p(t)=|u(t)s,r|2, and there is perfect state transfer if p(t0)=1p(t0)=1 for some t0t0. Under the hypothesis of perfect state transfer, we provide closed form expressions for dkpdtk at t0t0 for any k∈Nk∈N. Those expressions then yield an easily computed lower bound on p(t0+h)p(t0+h) for any h. We also produce expressions for the first two partial derivatives of p with respect to the weight of an edge in G, with the expression for the second derivative holding under the hypothesis of perfect state transfer. A parallel suite of results using the Laplacian matrix of G is also developed, and examples illustrating the results are included. The techniques rely on the spectral decomposition of the adjacency (respectively, Laplacian) matrix, and on perturbation theory for eigenvalues and eigenvectors of symmetric matrices.