量子系综的反馈镇定：复旗流形上的全局收敛性分析
Feedback stabilization of quantum ensembles: a global convergence analysis on complex flag manifolds
摘要：
在N能级量子力学系统中，混合密度算符的幺正反馈镇定到周期轨道的问题需要一个自然的基于李雅普诺夫的时变反馈设计。闭环系统的吸引域的全局描述可以基于密度算子空间的类似“根空间”的结构来提供。这个凸集作为一个复杂的旗形流形，其中每个叶子都与密度算子的本征值的共伴轨道相一致。收敛条件是时间无关的，但取决于从旗流形的拓扑结构：它表明，闭环必须有一些平衡至少等于欧拉特征的流形，从而施加障碍的拓扑性质的全局稳定性。
英文摘要：
In an N-level quantum mechanical system, the problem of unitary feedback stabilization of mixed density operators to periodic orbits admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a ``root-space''-like structure of the space of density operators. This convex set foliates as a complex flag manifold where each leaf is identified with the coadjoint orbit of the eigenvalues of the density operator. The converging conditions are time-independent but depend from the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing obstructions of topological nature to global stabilizability.
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