Algebraic conditions for indirect controllability in quantum coherent feedback schemes

In coherent feedback control schemes a target quantum system S is put in contact with an auxiliary system A and the coherent control can directly affect only A. The system S is controlled indirectly through the interaction with A. The system S is said to be indirectly controllable if every unitary transformation can be performed on the state of S with this scheme. In this paper we show how indirect controllability of S is equivalent to complete controllability of the combined system S + A, if the dimension of A is ≥ 3. In the case where the dimension of A is equal to 2, it is possible to have indirect controllability without having complete controllability of S +A and we give sufficient conditions for this to happen. We conjecture that these conditions are also necessary. The results of the paper extend the result of [5] and expand the results of [6] to systems of arbitrary dimensions.