Information transmission under Markovian noise

We consider an open quantum system undergoing Markovian dynamics, the latter being modelled by a discrete-time quantum Markov semigroup $({\mathrm{\Phi }}^n{)}_{n\in \mathbb{N}}$, resulting from the action of sequential uses of a quantum channel $\mathrm{\Phi }$, with $n\in \mathbb{N}$ being the discrete time parameter. We find upper and lower bounds on the one-shot $\epsilon$-error information transmission capacities of ${\mathrm{\Phi }}^n$ for a finite time $n\in \mathbb{N}$ and $\epsilon \in [0,1)$ in terms of the structure of the peripheral space of the channel $\mathrm{\Phi }$. We consider transmission of $(i)$ classical information (both in the unassisted and entanglement-assisted settings); $(ii)$ quantum information and $(iii)$ private classical information.