Note on the Controllability of Molecular Systems 1 a Slight Improvement

The purpose of this brief note is to show that the set of states reachable from a given initial state for a finite dimensional quantum system is equal to the orbit under the Lie group generated by the Lie algebra generated by the internal and external Hamiltonians, even if this Lie group is not compact. This result has been explicitly mentioned in several talks by the author. This note is not being submitted for publication in any journal. Consider a finite-dimensional quantum system: ˙ c = Ac + Bcǫ(t), c(0) = c 0 (1.1) Here c evolves on the complex sphere in some C N , and A and B are two skew-Hermitian matrices. It is well known that the set of states reachable from c 0 , via the application of admissible controls, equals the orbit through c 0 of the group, G corresponding to the Lie algebra generated by the matrices A and B, provided this Lie group is compact. See [5, 6, 4] for instance. This result is valid even if G is not compact. This result has been mentioned explicitly by the author in several talks and grant proposals, though the result has been recorded only cryptically in [7, 8] (primarily because it offers no significant practical improvements). Since such a nominal improvement may, nevertheless, be of interest, a discussion of it is provided below.