通过由Jacobi矩阵确定相互作用的自旋链的量子通信
Quantum communication through a spin chain with interaction determined by a Jacobi matrix
摘要：
我们得到了描述具有各向同性最近邻XY耦合的线性自旋链中单自旋激发态演化的含时关联函数，其中哈密顿量与一组正交多项式的Jacobi矩阵有关.对于Krawtchouk多项式的情况下，任意元素的关联函数表示在一个简单的封闭形式。它的渐近极限对应于Charlier多项式的Jacobi矩阵，可以理解为海森堡群元素的酉演化。本文还研究了哈恩多项式、对偶哈恩多项式和拉卡多项式的雅可比矩阵所对应的哈密顿量的关联函数。对于哈恩多项式，我们得到了一般的关联函数，它的一些特殊情况下，和极限相关的迈克斯纳多项式，其中su（1，1）代数描述的基本对称性。对于对偶Hahn多项式和Racah多项式的情况，关联函数的一般表达式包含非超几何型的求和。然而，在特殊情况下会出现简化。
英文摘要：
We obtain the time-dependent correlation function describing the evolution of a single spin excitation state in a linear spin chain with isotropic nearest-neighbour XY coupling, where the Hamiltonian is related to the Jacobi matrix of a set of orthogonal polynomials. For the Krawtchouk polynomial case an arbitrary element of the correlation function is expressed in a simple closed form. Its asymptotic limit corresponds to the Jacobi matrix of the Charlier polynomial, and may be understood as a unitary evolution resulting from a Heisenberg group element. Correlation functions for Hamiltonians corresponding to Jacobi matrices for the Hahn, dual Hahn and Racah polynomials are also studied. For the Hahn polynomials we obtain the general correlation function, some of its special cases, and the limit related to the Meixner polynomials, where the su(1,1) algebra describes the underlying symmetry. For the cases of dual Hahn and Racah polynomials the general expressions of the correlation functions contain summations which are not of hypergeometric type. Simplifications, however, occur in special cases.
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