On the problem of quantum control in infinite dimensions

TL;DR

This paper investigates the limitations and possibilities of quantum control in infinite-dimensional systems, showing that while exact control is impossible, dense control orbits can be achieved under certain conditions.

Contribution

It establishes the compatibility between the non-controllability of exact states and the density of reachable states in infinite-dimensional quantum systems.

Findings

The reachable set's complement is dense in certain function spaces.

Dense orbits can be generated by specific operators.

The closure of the reachable set is dense in the Hilbert space.

Abstract

In the framework of bilinear control of the Schr\"odinger equation with bounded control operators, it has been proved that the reachable set has a dense complemement in ${\cal S}\cap {\cal H}^{2}$. Hence, in this setting, exact quantum control in infinite dimensions is not possible. On the other hand it is known that there is a simple choice of operators which, when applied to an arbitrary state, generate dense orbits in Hilbert space. Compatibility of these two results is established in this paper and, in particular, it is proved that the closure of the reachable set of bilinear control by bounded operators is dense in ${\cal S}\cap {\cal H}^{2}$. The requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group.