群关联方案网络上量子点的完美状态转移
Perfect state transfer of a qudit over underlying networks of group association schemes
摘要：
作为Christandl等人的\cite{8,9""}和Facer等人的\cite{Facer}的结果的推广，Bernasconi等人的\cite{godsil,godsil1}研究了由一大类立方体图（例如，超立方体），它们是初等交换群$Z_2^n$的Cayley图。在参考文献中。\cite{PST,psd}、距离正则自旋网络上量子比特的PST和伪距离正则网络上$d$能级量子态的最优态转移（ST），其中网络一般不与某个有限群相关.本文研究了量子在更一般的网络（称为关联方案的底层网络）的对极上的PST。特别是，我们考虑群关联方案的底层网络，以便利用群的性质（如不可约字符），并使用这些网络的代数结构（例如Bose-Mesner代数）为了给出哈密顿量中的耦合常数的显式分析公式，使得最初在一个位点上编码的特定量子点的状态将完美地演化到相反的位点，动态控制它表明，唯一的必要条件，以PST在这些网络上实现的是，相应的组的中心是非平凡的。因此，所有具有非平凡中心的群，如交换群、具有偶数$n$的二面体群$D_{2n}$、Clifford群CL（n）和所有$p$ -群，都可以得到群结合方案的底层网络上的PST。
英文摘要：
As generalizations of results of Christandl et al.\cite{8,9""} and Facer et al.\cite{Facer}, Bernasconi et al.\cite{godsil,godsil1} studied perfect state transfer (PST) between two particles in quantum networks modeled by a large class of cubelike graphs (e.g., the hypercube) which are the Cayley graphs of the elementary abelian group $Z_2^n$. In Refs. \cite{PST,psd}, respectively, PST of a qubit over distance regular spin networks and optimal state transfer (ST) of a $d$-level quantum state (qudit) over pseudo distance regular networks were discussed, where the networks considered there were not in general related with a certain finite group. In this paper, PST of a qudit over antipodes of more general networks called underlying networks of association schemes, is investigated. In particular, we consider the underlying networks of group association schemes in order to employ the group properties (such as irreducible characters) and use the algebraic structure of these networks (such as Bose-Mesner algebra) in order to give an explicit analytical formula for coupling constants in the Hamiltonians so that the state of a particular qudit initially encoded on one site will perfectly evolve to the opposite site without any dynamical control. It is shown that the only necessary condition in order to PST over these networks be achieved is that the centers of the corresponding groups be non-trivial. Therefore, PST over the underlying networks of the group association schemes over all the groups with non-trivial centers such as the abelian groups, the dihedral group $D_{2n}$ with even $n$, the Clifford group CL(n) and all of the $p$-groups can be achieved.
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