  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. We extend our previous result on nonexistence of "truly quantum"
protocols that protect an unknown qubit state against the depolarizing noise
better than "classical" ones [Phys. Rev. A, 95, 022321 (2017)] in two
directions. First, we show that the statement is also true in any
finite-dimensional Hilbert spaces, which was previously conjectured, the
optimal protocol is either the do nothing protocol or the discriminate and
reprepare protocol, depending on the strength of the noise. Second, in the case
of a qubit, we show that essentially the same conclusion holds for any unital
noise. These results describe the fundamental limitations in quantum mechanics
from the viewpoint of control theory.
