Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems

TL;DR

This paper establishes spectral conditions involving conical eigenvalue intersections that ensure controllability of quantum systems, linking spectral properties to Lie algebraic controllability and demonstrating equivalence of approximate and exact controllability in finite dimensions.

Contribution

It introduces spectral conditions based on conical eigenvalue intersections that guarantee controllability and connects spectral properties with Lie algebraic controllability in quantum systems.

Findings

Spectral conditions for controllability are sufficient and structurally stable.

Approximate controllability in infinite dimensions is achieved under these spectral conditions.

In finite dimensions, spectral conditions imply exact controllability and Lie algebra generation.

Abstract

We study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in the case of three controls or in the case of two controls when the Hamiltonian is real. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the finite-dimensional case it implies that the system is Lie-bracket generating when lifted to the group of unitary transformations, and in particular that it is exactly controllable. Hence, Lie algebraic conditions are deduced from purely spectral properties. We conclude the article by proving that approximate and exact controllability are equivalent properties for general finite-dimensional quantum systems.