Uncontrollable quantum systems: A classification scheme based on Lie subalgebras

It is well known that a finite level quantum system is controllable if and only if the Lie algebra of its generators has full rank. When the rank of the Lie algebra is not full, there is a rich mathematical and physical structure to the subalgebra that to date has been analyzed only in special cases. We show that uncontrollable systems can be classified into reducible and irreducible ones. The irreducible class is the more subtle and can be related to a notion of generalized entanglement. We give a general prescription for revealing irreducible uncontrollable systems: the fundamental representation of su(N), where N is the number of levels, must remain irreducible in the subalgebra of su(N). We illustrate the concepts with a variety of physical examples.