Dynamic homotopy and landscape dynamical set topology in quantum control

Jason Dominy, Herschel Rabitz

Research output: Contribution to journalArticle

4 Scopus citations

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 |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.

Original languageEnglish (US)
Article number082201
JournalJournal of Mathematical Physics
Volume53
Issue number8
DOIs
StatePublished - Aug 1 2012

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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