Abstract

Hamiltonian symmetry and controllability are found to be closely related in quantum control theory. For spin chains with XXZ coupling subject to a local control in Z direction on the kth spin, the total Hilbert space is decomposed into series of invariant excitation subspaces under the dynamics. We investigate which locations of the local Z control can induce controllability on the first excitation subspace. It is found that the uncontrollable cases are one-to-one correspondent with the Hamiltonian external symmetries. We also use Bethe ansatz to find all such symmetries of XXZ chain. Among different local control strategies, we concentrate on local end control and derive the subspace controllability. In addition, we investigate what are the minimal control resources for a fully controllable XYZ chain, and find the controllability proof for different control strategies. Some of the results have been included in http://arxiv.org/abs/1012.3695