  Consider an open quantum system with (discrete-time) Markovian dynamics. Our
task is to store information in the system in such a way that it can be
retrieved perfectly, even after the system is left to evolve for an arbitrarily
long time. We show that this is impossible for classical (resp. quantum)
information precisely when the dynamics is mixing (resp. asymptotically
entanglement breaking). Furthermore, we provide tight universal upper bounds on
the minimum time after which any such dynamics `scrambles' the encoded
information beyond the point of perfect retrieval. On the other hand, for
dynamics that are not of this kind, we show that information must be encoded
inside the peripheral space associated with the dynamics in order for it to be
perfectly recoverable at any time in the future. This allows us to derive
explicit formulas for the maximum amount of information that can be protected
from noise in terms of the structure of the peripheral space of the dynamics.
