  In quantum control theory, the fundamental issue of controllability covers
the questions whether and under which conditions a system can be steered from
one pure state into another by suitably tuned time evolution operators. Even
though Lie theoretic methods to analyze these aspects are well-established for
finite dimensional systems, they fail to apply to those with an infinite number
of levels. The Jaynes-Cummings-Hubbard model - describing two-level systems in
coupled cavities - is such an infinite dimensional system. In this contribution
we study its controllability. In the two cavity case we exploit symmetry
arguments; we show that one part of the control Hamiltonians can be studied in
terms of infinite dimensional block diagonal Lie algebras while the other part
breaks this symmetry to achieve controllability. An induction on the number of
cavities extends this result to the general case. Individual control of the
qubit and collective control of the hopping between cavities is sufficient for
both pure state and strong operator controllability. We additionally establish
new criteria for the controllability of infinite dimensional quantum systems
admitting symmetries.
