Approaches for approximate additivity of the Holevo information of quantum channels

We study quantum channels that are close to another channel with weakly additive Holevo information, and we derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above, which measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as well as an outer bound on the dynamic capacity region of a quantum channel. Since these parameters are defined in terms of the diamond distance, the upper bounds can be computed efficiently using semidefinite programming (SDP). We exhibit the usefulness of our method with two example channels: a convex mixture of amplitude damping and depolarizing noise and a composition of amplitude damping and dephasing noise. For both channels, our bounds perform well in certain regimes of the noise parameters in comparison to a recently derived SDP upper bound on the classical capacity. Along the way, we define the notion of a generalized channel divergence (which includes the diamond distance as an example), and we prove that for jointly covariant channels these quantities are maximized by purifications of a state invariant under the covariance group. This latter result may be of independent interest.

Figure 1

Plot of the Hadamard parameters of the amplitude damping channel ${\mathcal{A}}_p$. The Hadamard parameter ${Had}_{\mathcal{S}}\left({\mathcal{A}}_p\right)$ with $\mathcal{S}=\left\{\right|0\rangle ,|1\rangle \}$, as given in Eq. (14), is plotted in blue/dashed, and the Hadamard parameter ${Had}_{\mathrm{deg}}\left({\mathcal{A}}_p\right)$, as given in Definition III.12, is plotted in red/dash-dotted.

Figure 2

Lower and upper bound on the classical capacity $C\left({\mathcal{A}}_p\right)$ of the amplitude damping channel ${\mathcal{A}}_p$ defined in Eq. (28). The Holevo information $\chi \left({\mathcal{A}}_p\right)$, a lower bound on $C\left({\mathcal{A}}_p\right)$, is plotted in black/dashed, and the strong converse upper bound $C_{\beta }\left({\mathcal{A}}_p\right)$, defined in Eq. (30) and equal to $log(1+\sqrt{1-p})$, is plotted in blue/solid.

Figure 3

Upper and lower bounds on the classical capacity $C\left({\mathcal{N}}_p\right)$ of the channel ${\mathcal{N}}_p$ defined in Eq. (32). The Holevo information $\chi \left({\mathcal{N}}_p\right)$, a lower bound on $C\left({\mathcal{N}}_p\right)$, is plotted in black/dashed, the upper bound $\chi \left({\mathcal{N}}_p\right)+2{\varepsilon }_p+g\left({\varepsilon }_p\right)$ on $C\left({\mathcal{N}}_p\right)$ from Corollary III.5(i) with the covariance parameter ${\varepsilon }_p:={cov}_{\mathcal{P}}\left({\mathcal{N}}_p\right)=\frac{1}{2}{p}^2$ is plotted in blue/solid, and the strong converse upper bound $C_{\beta }\left({\mathcal{N}}_p\right)$ defined in Eq. (30) is plotted in red/dash-dotted.

Figure 4

Plot of the entanglement-breaking parameter $EB\left({\mathcal{M}}_p\right)$ of the channel ${\mathcal{M}}_p$ defined in Eq. (33) for the interval $p\in [0,1]$ (left) and zoomed in on the interval $p\in [0.35,0.75]$ (right).

Figure 5

Upper and lower bounds on the classical capacity $C\left({\mathcal{M}}_p\right)$ of the channel ${\mathcal{M}}_p$ defined in Eq. (33). The Holevo information $\chi \left({\mathcal{M}}_p\right)$ is plotted in black/dashed, the upper bound $\chi \left({\mathcal{M}}_p\right)+3{\varepsilon }_p+2g\left({\varepsilon }_p\right)$ on $C\left({\mathcal{M}}_p\right)$ from Corollary III.7 with the entanglement-breaking parameter ${\varepsilon }_p:=EB\left({\mathcal{M}}_p\right)$ is plotted in green/dotted, the upper bound $\chi \left({\mathcal{M}}_p^{\mathrm{EB}}\right)+2{\varepsilon }_p+g\left({\varepsilon }_p\right)$ from Corollary III.7 is plotted in blue/solid, and the strong converse upper bound $C_{\beta }\left({\mathcal{M}}_p\right)$ defined in Eq. (30) is plotted in red/dash-dotted. The channel ${\mathcal{M}}_p^{\mathrm{EB}}$ is the entanglement-breaking channel closest in diamond distance to ${\mathcal{M}}_p$ found by the SDP in Eq. (10).