2011/04/11 by Bašić, Milan
#Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Physical sciences #Information Theory (cs.IT) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.1104.1825
In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph G, which is characterized by its circulant adjacency matrix A. Formally, we say that there exists a \it perfect state transfer (PST) between vertices a,b∈ V(G) if |F(τ)ab|=1, for some positive real number τ, where F(t)=exp(ıAt). Saxena, Severini and Shparlinski (\it International Journal of Quantum Information 5 (2007), 417--430) proved that |F(τ)aa|=1 for some a∈ V(G) and τ∈ \R+ if and only if all eigenvalues of G are integer (that is, the graph is integral). The integral circulant graph \ICGn (D) has the vertex set Zn = \0, 1, 2, ..., n - 1\ and vertices a and b are adjacent if gcd(a-b,n)∈ D, where D ⊆ \d : d | n, 1≤ d