  Given a quantum Markovian noise model, we study the maximum dimension of a
classical or quantum system that can be stored for arbitrarily large time. We
show that, unlike the fixed time setting, in the limit of infinite time, the
classical and quantum capacities are characterized by efficiently computable
properties of the peripheral spectrum of the quantum channel. In addition, the
capacities are additive under tensor product, which implies in the language of
Shannon theory that the one-shot and the asymptotic i.i.d. capacities are the
same. We also provide an improved algorithm for computing the structure of the
peripheral subspace of a quantum channel, which might be of independent
interest.
