量子控制中的动态同伦与景观动力学集合拓扑
Dynamic Homotopy and Landscape Dynamical Set Topology in Quantum Control
AI总结：
通过证明端点映射是Hurewicz纤维化，揭示了将初始态驱动到目标态的控制子集与状态流形的环路空间同伦等价，从而提供了控制空间子集的丰富拓扑信息。
摘要：
我们研究了量子控制中将给定初始态驱动到给定最终态的控制子集的拓扑，其中“态”可能指纯态|ψ>、系综密度矩阵ρ或酉传播子U(0,T)。分析表明，对于一类具有向量控制的仿射控制系统，作用于控制空间的端点映射是Hurewicz纤维化。利用由此产生的纤维化序列和保持基点的映射同伦类的长正合序列，我们证明了所指示的控制子集与状态流形的环路空间同伦等价。这不仅使我们能够理解通过该端点映射实现的状态空间子集的原像“动力学集合”的连通性，而且为控制空间的此类子集提供了丰富的额外拓扑信息。
英文摘要：
We examine the topology of the subset of controls taking a given initial state to a given final state in quantum control, where "state" may mean a pure state |ψ>, an ensemble density matrix ρ, or a unitary propagator U(0,T). The analysis consists in showing that the endpoint map acting on control space is a Hurewicz fibration for a large class of affine control systems with vector controls. Exploiting the resulting fibration sequence and the long exact sequence of basepoint-preserving homotopy classes of maps, we show that the indicated subset of controls is homotopy equivalent to the loopspace of the state manifold. This not only allows us to understand the connectedness of "dynamical sets" realized as preimages of subsets of the state space through this endpoint map, but also provides a wealth of additional topological information about such subsets of control space.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
