EXACT ALGEBRAIC CONDITIONS FOR INDIRECT CONTROLLABILITY OF QUANTUM SYSTEMS

In several quantum 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 that is not coupled 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(S)n(A)) or L = u(n(S)n(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 of 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.