Complete controllability is a fundamental issue in the field of control of
quantum systems, not least because of its implications for dynamical realiz
ability of the kinematical bounds on the optimization of observables. In th
is paper we investigate the question of complete controllability for finite
-level quantum systems subject to a single control field, for which the int
eraction is of dipole form. Sufficient criteria for complete controllabilit
y of a wide range of finite-level quantum systems are established and the q
uestion of limits of complete controllability is addressed. Finally, the re
sults are applied to give a classification of complete controllability for
four-level systems.