自旋量子系统的可控李代数计算

Calculations of the controllability Lie algebra for spin quantum systems

动态系统可控性同是经典和量子控制中研究的一个基本问题,本文研究了单自旋和双自旋量子系统的可控李代数的计算.首先基于量子系统可控的等价性条件,通过单量子系统Hamiltonian算符的李括号运算,给出了与基系数相关的系统可控的充要条件.然后利用Cartan分解方法构造了李代数su(4)的矩阵基,同时根据可控性基本定理提出了Hamiltonian算符多重李括号的计算方法及系统的可控性判据.

Controllability of dynamical systems is a fundamental problem for both classical control as well as quantum control. This paper focuses on investigating the controllability Lie algebra of single-spin and two spin quantum systems. Firstly, based on the equivalent controllability conditions of quantum systems, some general necessary and sufficient conditions related to the basis coefficients are obtained by computing Lie brackets of Hamiltonian operators for single-spin quantum systems. Secondly, the matrix basis of Lie algebra su（4） is generated by Cartan decomposition. Using the matrix basis and employing the controllability theorem of quantum system, a calculation method of multiple Lie brackets for the Hamiltonian operators and a controllability criterion of quantum systems are also put forward in this article.