Fast convergence of Dynamic Capacities of GNS-Symmetric Quantum Channels

We consider a quantum system described by a quantum channel $\mathrm{\Phi }$ that is applied at every time step and study the time evolution of its information capacities. When $\mathrm{\Phi }$ is a GNS-symmetric channel (this includes Pauli channels, for example), we give explicit exponential convergence bounds for the classical and quantum capacities. These bounds are in terms of entropic properties of $\mathrm{\Phi }$. We further illustrate how these results help quantify the performance of active versus passive error-correction setups.