Regularized maximum pure-state input-output fidelity of a quantum channel

As a toy model for the capacity problem in quantum information theory we investigate finite and asymptotic regularizations of the maximum pure-state input-output fidelity $F(\cal N$) of a general quantum channel $\cal N$. We show that the asymptotic regularization $\tilde F(\cal N$) is lower bounded by the maximum output $\infty$-norm $\nu_\infty(\cal N)$ of the channel. For $\cal N$ being a Pauli channel we find that both quantities are equal.