Noise suppression by quantum control before and after the noise

We discuss the possibility of protecting the state of a quantum system that goes through noise by measurements and operations before and after the noise process. The aim is to seek the optimal protocol that makes the input and output states as close as possible and to clarify the role of the measurements therein. We consider two cases: one can perform quantum measurements and operations (i) only after the noise process and (ii) both before and after. We prove in a two-dimensional Hilbert space that, in case (i), the noise suppression is essentially impossible for all types of noise and, in case (ii), the optimal protocol for the depolarizing noise is either the "do nothing" protocol or the "discriminate and reprepare" protocol. These protocols are not "truly quantum" and can be considered as classical. They involve no measurement or only use the measurement outcomes. These results describe the fundamental limitations in quantum mechanics from the viewpoint of control theory. Finally, we conjecture that a statement similar to case (ii) holds for higher-dimensional Hilbert spaces and present some numerical evidence.