  A new notion of controllability for quantum systems that takes advantage of
the linear superposition of quantum states is introduced. We call such notion
von Neumann controllabilty and it is shown that it is strictly weaker than the
usual notion of pure state and operator controlability. We provide a simple and
effective characterization of it by using tools from the theory of unitary
representations of Lie groups. In this sense we are able to approach the
problem of control of quantum states from a new perspective, that of the theory
of unitary representations of Lie groups. A few examples of physical interest
and the particular instances of compact and nilpotent dynamical Lie groups are
discussed.
