The Controllability of Infinite Quantum Systems and Closed Subspace Criteria

Quantum phenomena of interest in connection with quantum computation and communication often deal with transfers between eigenstates, and their linear superpositions. For systems having only a finite number of states, the quantum evolution equation (the Schrödinger equation) is finite-dimensional and the results on controllability on Lie groups as worked out decades ago [1] provide most of what is needed insofar as controllability of non-dissipative systems is concerned. However, for infinite-dimensional evolution of quantum systems, many difficulties, both conceptual and technical, remain. In this paper we organize some recent results from the physics literature in control-theoretic terms and emphasize the type of analysis needed to go beyond what basic differential geometry can provide. In particular, we analyze the problem of controllability of quantum systems subject to the constraint that the trajectories must lie in pre-defined subspaces, and discuss from a controllability viewpoint the important results of Law and Eberly [2].