  Controllability -- the possibility of performing any target dynamics by
applying a set of available operations -- is a fundamental requirement for the
practical use of any physical system. For finite-dimensional systems, as for
instance spin systems, precise criterions to establish controllability, such as
the so called rank criterion, are well known. However most physical systems
require a description in terms of an infinite-dimensional Hilbert space whose
controllability properties are poorly understood. Here, we investigate
infinite-dimensional bosonic quantum systems -- encompassing quantum light,
ensembles of bosonic atoms, motional degrees of freedom of ions, and
nano-mechanical oscillators -- governed by quadratic Hamiltonians (such that
their evolution is analogous to coupled harmonic oscillators). After having
highlighted the intimate connection between controllability and recurrence in
the Hilbert space, we prove that, for coupled oscillators, a simple extra
condition has to be fulfilled to extend the rank criterion to infinite
dimensional quadratic systems. Further, we present a useful application of our
finding, by proving indirect controllability of a chain of harmonic
oscillators.
