Dynamic homotopy and landscape dynamical set topology in quantum control
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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Aug 01, 2012
- Source ID
- 10.1063/1.4742375
Entities
People
- Herschel A. Rabitz
- Jason Dominy
Organizations
- Army Research Office
- National Science Foundation
- Princeton University
- United States Department of Energy