Control of a two-level quantum system in a coherent feedback scheme

We consider a scheme for the control of a spin 1/2, S, where an auxiliary spin 1/2, A, plays the role of the controller, and an Ising interaction is assumed between the spins. We demonstrate that even if there is no complete controllability on the system S + A we can have full control on the system S if and only if the initial state of A is pure. This provides the simplest example where the system S is indirectly controllable while the whole system S + A is not completely controllable. We also give an explicit algorithm for the indirect control of S.