Dynamic Homotopy and Landscape Dynamical Set Topology in Quantum Control

TL;DR

This paper explores the topological structure of control sets in quantum control, revealing their homotopy equivalence to loopspaces and providing insights into their connectedness and topology.

Contribution

It demonstrates that the endpoint map in quantum control is a Hurewicz fibration, leading to new topological understanding of control subsets via homotopy theory.

Findings

Control subsets are homotopy equivalent to loopspaces of the state manifold.

The endpoint map acts as a Hurewicz fibration for a broad class of control systems.

Provides topological insights into the connectedness of dynamical control sets.

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 |\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.