2012/06/01 by Xiaoxia Fan, Chris Godsil, Fan, Xiaoxia +1 · 1 citation
Chemistry · Computer Science · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Molecular spectroscopy and chirality #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1206.0082
openalex publication_date 2012/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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). Physicists 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, \normγ=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 \normU(t)eu - γev < ε. Again we necessarily have |γ|=1. 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 study double-star graphs, which are trees with two vertices of degree k+1 and all other vertices with degree one. We prove that there is never perfect state transfer between the two vertices of degree k+1, and that there is pretty good state transfer between them if and only if 4k+1 is a perfect square.