  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.
