具有完美状态转移的循环图的特征
Characterization of circulant graphs having perfect state transfer
摘要：
在本文中，我们回答了具有最近邻耦合的循环量子自旋网络何时可以实现完美态转移的问题。该网络由循环图$G$描述，其特征在于其循环邻接矩阵$A$。形式上，我们说，如果$|F(τ)_{ab}|=1$，对于某个正实数$τ$，其中$F(t)=\exp(ıAt)$，在顶点$a,b\in V(G)$之间存在{ \it完美状态转移}（PST）。Saxena、Severini和Shparlinski（{ \it International Journal of Quantum Information} 5（2007），417- 430）证明了对于某些$a\in V(G)$和$τ\in \R^+$，$|F(τ)_{aa}|=1$当且仅当$G$的所有特征值都是整数（即图是整数）。整数循环图$\ICG_n (D)$具有顶点集$Z_n = \{0, 1, 2, ..., n - 1\}$，如果$\gcd(a-b,n)\in D$，则顶点$a$和$b$相邻，其中$D \subseteq \{d : d \mid n,\ 1\leq d<n\}$。这些图是高度对称的，在化学图论中有重要的应用。我们证明了$\ICG_n (D)$有PST当且仅当$n\in 4\N$和$D=\widetilde{D_3}\cup D_2\cup 2D_2\cup 4D_2\cup \{n/2^a\}$，其中$\widetilde{D_3}=\{d\in D\ |\ n/d\in 8\N\}$，$D_2= \{d\in D\ |\ n/d\in 8\N+4\}\setminus \{n/4\}$和$a\in\{1,2\}$。因此，我们已经回答了Angeles-Canul { \it et al.}在{ \it Quantum Information and Computation}，Vol.10，No.3&4（2010）0325- 0342中提出的积分循环图中的完美状态转移的完全表征的问题。此外，我们还计算了完美量子通信距离（PST出现的顶点之间的距离），并描述了具有PST的整数循环图的谱。最后给出了一个计算给定阶的具有PST的整循环图个数的封闭表达式。
英文摘要：
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 $a,b\in V(G)$ 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 $a\in V(G)$ and $τ\in \R^+$ if and only if all eigenvalues of $G$ are integer (that is, the graph is integral). The integral circulant graph $\ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, ..., n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, where $D \subseteq \{d : d \mid n,\ 1\leq d<n\}$. These graphs are highly symmetric and have important applications in chemical graph theory. We show that $\ICG_n (D)$ has PST if and only if $n\in 4\N$ and $D=\widetilde{D_3}\cup D_2\cup 2D_2\cup 4D_2\cup \{n/2^a\}$, where $\widetilde{D_3}=\{d\in D\ |\ n/d\in 8\N\}$, $D_2= \{d\in D\ |\ n/d\in 8\N+4\}\setminus \{n/4\}$ and $a\in\{1,2\}$. We have thus answered the question of complete characterization of perfect state transfer in integral circulant graphs raised in {\it Quantum Information and Computation}, Vol. 10, No. 3&4 (2010) 0325--0342 by Angeles-Canul {\it et al.} Furthermore, we also calculate perfect quantum communication distance (distance between vertices where PST occurs) and describe the spectra of integral circulant graphs having PST. We conclude by giving a closed form expression calculating the number of integral circulant graphs of a given order having PST.
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