Feedback stabilization of quantum ensembles: a global convergence analysis on complex flag manifolds

TL;DR

This paper analyzes the global convergence of feedback stabilization in quantum ensembles, revealing topological obstructions related to complex flag manifolds that limit the possibility of achieving global stabilization.

Contribution

It provides a novel topological analysis of feedback stabilization in quantum systems using complex flag manifolds and identifies fundamental obstructions to global stabilization.

Findings

The domain of attraction can be described using a root-space structure on density operators.

The flag manifold's topology imposes lower bounds on the number of equilibria.

Topological obstructions prevent global stabilization in certain quantum systems.

Abstract

In an N-level quantum mechanical system, the problem of unitary feedback stabilization of mixed density operators to periodic orbits admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a ``root-space''-like structure of the space of density operators. This convex set foliates as a complex flag manifold where each leaf is identified with the coadjoint orbit of the eigenvalues of the density operator. The converging conditions are time-independent but depend from the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing obstructions of topological nature to global stabilizability.