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

Figure 1

Regularized maximum input-output fidelity for a Pauli channel $\mathcal{P}$ with probabilities $p=(0.1,0.2,0.3,0.4)$: $F^{\left(n\right)}\left(\mathcal{P}\right)$ determined by numerical maximization, $\diamond$; $F{({\mathcal{P}}^{\otimes n},{\psi }_n)}^{1/n}$ for trial states ${\psi }_n, +$; and $\tilde{F}\left(\mathcal{P}\right)$, - - -.

Figure 2

Regularized maximum input-output fidelity for a Pauli channel with probabilities $p=(0.286,0,0.333,0.381)$. Symbols have the same meaning as in Fig. 1.

Figure 3

Regularized maximum input-output fidelity for a Pauli channel with probabilities $p=(0.308,0,0.333,0.358)$. Symbols have the same meaning as in Fig. 1.