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 $F(\mathcal{N}$) of a general quantum channel $\mathcal{N}$. We show that the asymptotic regularization $\tilde{F}(\mathcal{N}$) is lower bounded by the maximum output $\mathrm{\infty }$-norm ${\nu }_{\mathrm{\infty }}(\mathcal{N})$ of the channel. For $\cal N$ being a Pauli channel we find that both quantities are equal.