Further Results on Stabilizing Control of Quantum Systems

It is notoriously known that in contrast to the measure of classical macro systems, the quantum measurement will in general cause quantum state collapse. In this note, we will present some further results on stabilizing control of quantum systems with measurement being involved. To be precise, we will study the effects of the open-loop control and the feedback control in preparing an arbitrarily desired eigenstate of an effective Hamiltonian, respectively. For the open-loop control case, it is shown that no matter how to select the measurement channel, control channel and the control law, one cannot prepare all the eigenstates of the effective Hamiltonian with arbitrary high fidelity for all sufficient large time. While for the feedback case, we not only demonstrate how to select the measurement channel properly to achieve the control target, but also show if not, what cannot be done by the measurement-based feedback control.