Universal control of quantum subspaces and subsystems

We describe a broad dynamical-algebraic framework for analyzing the quantum control properties of a set of naturally available interactions. General conditions under which universal control is achieved over a set of subspaces/subsystems are found in terms of the representation theory of the group U_A of allowed quantum evolutions. In some cases universal control can be achieved in all the state-space sectors carrying an irreducible action of U_A. All known physical examples of universal control on subspaces/systems are related to the framework developed here. Implications for quantum-information processing are discussed.