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Dynamic Homotopy and Landscape Dynamical Set Topology in Quantum Control

By Jason Dominy and Herschel Rabitz


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 |\psi>, an ensemble density matrix \rho, 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.Comment: Minor clarifications, and added new appendix addressing scalar control of 2-level quantum system

Topics: Quantum Physics, Computer Science - Systems and Control, Mathematics - Optimization and Control, 81Q93, 55P05, 55P10
Year: 2012
DOI identifier: 10.1063/1.4742375
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