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# Dynamic homotopy and landscape dynamical set topology in quantum control

## Author(s): Dominy, Jason; Rabitz, Herschel

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 Abstract: 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 vertical bar 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. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4742375] Publication Date: Aug-2012 Electronic Publication Date: 13-Aug-2012 Citation: Dominy, Jason, Rabitz, Herschel. (2012). Dynamic homotopy and landscape dynamical set topology in quantum control. JOURNAL OF MATHEMATICAL PHYSICS, 53 (10.1063/1.4742375 DOI: doi:10.1063/1.4742375 ISSN: 0022-2488 Pages: 082201-1 -082201-18 Type of Material: Journal Article Journal/Proceeding Title: JOURNAL OF MATHEMATICAL PHYSICS Version: Final published version. Article is made available in OAR by the publisher's permission or policy.

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