Pretty good quantum state transfer in asymmetric graphs via potential

Abstract

We construct infinite families of graphs in which pretty good state transfer can be induced by adding a potential to the nodes of the graph (i.e. adding a number to a diagonal entry of the adjacency matrix). Indeed, we show that given any graph with a pair of cospectral nodes, a simple modification of the graph, along with a suitable potential, yields pretty good state transfer (i.e. asymptotically perfect state transfer) between the nodes. This generalizes previous work, concerning graphs with an involution, to asymmetric graphs.where X i , Y i , Z i are the standard Pauli matrices, J ij denotes the strength of the XX coupling between node i and j, and the Q i 's give the strength of the magnetic field yielding an energy potential at each node.

An infinite family of circulant graphs with perfect state transfer in discrete quantum walks

“…However, if we are interested in pretty good state transfer from u to v, that is, for any ǫ > 0, there is a time k at which the probability of being on v is ǫ-close to 1, given initial state e u ⊗ x, then the eigenvalue conditions in Theorem 5.3 can be relaxed. Pretty good state transfer has been found in continuous quantum walks, see for example [5,24,15,17]. We would like to see discrete analogues.…”

Designing pretty good state transfer via isospectral reductions

“…is fulfilled for all λ. In words, u and v are cospectral if and only if, within each degenerate subspace, the sum of squares of absolute values of projections on sites u is equal to that of projections on site v. If u and v are cospectral and additionally [11] λ i |u = ± λ i |v…”

“…On the other hand, meeting the spectral requirements remains a difficult task. Nevertheless, in a recent paper [11] by Eisenberg et al, this has been rendered simpler for the case of PGST. They showed that Eqs.…”

“…More specifically, given a Hamiltonian H with two strongly cospectral sites u and v, its characteristic polynomial P can be decomposed [11] as…”

“…Such a method has been presented in Ref. [11], where several such mechanisms have been shown. In particular, the method shown there starts from a graph with cospectral vertices and selectively adds transcendental numbers to some diagonal entries of H, such that the modified setup features PGST.…”