Quantum control in infinite dimensions and Banach-Lie algebras: Pure point spectrum

TL;DR

This paper extends controllability criteria from finite to infinite-dimensional quantum systems using Banach-Lie algebra techniques, demonstrating approximate controllability with new methods and simpler proofs.

Contribution

It introduces an approximate Lie algebra rank condition for infinite-dimensional quantum control systems with unbounded drift Hamiltonians, simplifying controllability analysis.

Findings

Develops a scheme using recurrence and von Neumann algebras for approximate controllability.

Recovers previous genericity results with simpler proofs.

Provides examples demonstrating the effectiveness of the approach.

Abstract

In finite dimensions, controllability of bilinear quantum control systems can be decided quite easily in terms of the "Lie algebra rank condition" (LARC), such that only the systems Lie algebra has to be determined from a set of generators. In this paper we study how this idea can be lifted to infinite dimensions. To this end we look at control systems on an infinite dimensional Hilbert space which are given by an unbounded drift Hamiltonian and bounded control Hamiltonians. The drift is assumed to have empty continuous spectrum. We use recurrence methods and the theory of Abelian von Neumann algebras to develop a scheme, which allows us to use an approximate version of LARC, in order to check approximate controllability of the control system in question. Its power is demonstrated by looking at some examples. We recover in particular previous genericity results with a much easier proof. Finally several possible generalizations are outlined.