Equivalence between indirect controllability and complete controllability for quantum systems

We consider a control scheme where a quantum system S is put in contact with an auxiliary quantum system A and the control can affect A only, while S is the system of interest. The systems is then controlled indirectly through the interaction with A. Complete controllability of S + A means that every unitary state transformation for the system S + A can be achieved with this scheme. Indirect controllability means that every unitary transformation on the systems can be achieved. We prove in this paper, under appropriate conditions and definitions, that these two notions are equivalent in finite dimension. We use Lie algebraic methods to prove this result. (c) 2012 Elsevier B.V. All rights reserved.