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Exact Algebraic Conditions for Indirect Controllability in Quantum Coherent Feedback Schemes

In coherent quantum 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. The indirect controllability of S will depend on the `dynamical Lie algebra' L characterizing the dynamics of the total system S+A and on the initial state of the auxiliary system A. In this paper we describe this characterization exactly. A natural assumption is that the auxiliary system A is minimal which means that there is no part of A which is uncoupled to S, and we denote by n_A the dimension of such a minimal A, which we assume to be fully controllable. We show that, if n_A is greater than or equal to 3, indirect controllability of S is verified if and only if complete controllability of the total system S+A is verified, i.e., L=su(n_Sn_A) or L=u(n_Sn_A), where n_S denotes the dimension of the system S. If n_A=2, it is possible to have indirect controllability without having complete controllability. The exact condition for that to happen is given in terms of a Lie algebra L_S which describes the evolution on the system S only. We prove that indirect controllability is verified if and only if L_S=u(n_S), and the initial state of the auxiliary system A is pure.