  Manipulation of infinite dimensional quantum systems is important to
controlling complex quantum dynamics with many practical physical and chemical
backgrounds. In this paper, a general investigation is casted to the
controllability problem of quantum systems evolving on infinite dimensional
manifolds. Recognizing that such problems are related with infinite dimensional
controllability algebras, we introduce an algebraic mathematical framework to
describe quantum control systems possessing such controllability algebras. Then
we present the concept of smooth controllability on infinite dimensional
manifolds, and draw the main result on approximate strong smooth
controllability. This is a nontrivial extension of the existing controllability
results based on the analysis over finite dimensional vector spaces to analysis
over infinite dimensional manifolds. It also opens up many interesting problems
for future studies.
