Universal state transfer on graphs

Abstract

A continuous-time quantum walk on a graph G is given by the unitary matrix U (t) = exp(−itA), where A is the Hermitian adjacency matrix of G. We say G has pretty good state transfer between vertices a and b if for any ǫ > 0, there is a time t, where the (a, b)-entry of U (t) satisfies |U (t) a,b | ≥ 1 − ǫ. This notion was introduced by . The state transfer is perfect if the above holds for ǫ = 0. In this work, we study a natural extension of this notion called universal state transfer. Here, state transfer exists between every pair of vertices of the graph. We prove the following results about graphs with this stronger property:• Graphs with universal state transfer have distinct eigenvalues and flat eigenbasis (where each eigenvector has entries which are equal in magnitude).• The switching automorphism group of a graph with universal state transfer is abelian and its order divides the size of the graph. Moreover, if the state transfer is perfect, then the switching automorphism group is cyclic.• There is a family of prime-length cycles with complex weights which has universal pretty good state transfer. This provides a concrete example of an infinite family of graphs with the universal property.• There exists a class of graphs with real symmetric adjacency matrices which has universal pretty good state transfer. In contrast, Kay (2011) proved that no graph with real-valued adjacency matrix can have universal perfect state transfer.We also provide a spectral characterization of universal perfect state transfer graphs that are switching equivalent to circulants.

Universality in perfect state transfer

“…In this work, we extend some of the observations from Cameron et al [3]. More specifically, we prove new characterizations of graphs with universal perfect state transfer.…”

“…They showed that such graphs must have distinct eigenvalues, their unitary diagonalizing matrices must be type-II (see Chan and Godsil [4]), and their switching automorphism group must be cyclic. A spectral characterization for circulants with the universal property was also proved in [3].…”

“…et al [3] showed that it is possible to violate this "monogamy" property in graphs with complex Hermitian adjacency matrices. They studied graphs with a universal property where perfect state transfer occurs between every pair of vertices.…”

Pretty good state transfer of entangled states through quantum spin chains

“…Under the single-excitation framework, the state vectors 〉 j | are equivalent to the n vectors of the usual basis of the space  n , and the unitary operator = − U t ( ) e Ht i is equivalent to an × n n continuous-time quantum walk on the uniformly coupled path graph. This enables us to undertake a graph-theoretic approach in the study of PGST, as in [23,30]. The following theorem is the main result of [23]: Theorem 1.…”

“…We will now invoke a particular case of a theorem which was proved by Cameron et al (see theorem 3 in [30]). Theorem 2.…”