Single-letter quantum and private capacities of degradable channels

The tensor power of a degradable channel is degradable: the product degrading map ${\mathcal{D}}^{\otimes n}$ is a channel by Finite-dimensional channels compose and tensor, and it maps $({\mathcal{N}}^{\otimes n})(\rho )$ to $({\mathcal{N}}^c{)}^{\otimes n}(\rho )$. Therefore [F2] gives ${\sup }_{\rho }{I}_c(\rho ,{\mathcal{N}}^{\otimes n})=n{\max }_{\rho }{I}_c(\rho ,\mathcal{N})$. Substituting in the regularized quantum formula [F4] yields $Q(\mathcal{N})={\max }_{\rho }{I}_c(\rho ,\mathcal{N})$.

For any $n$-use input ensemble $\{p_x,{\rho }_x\}$, its tensor-power channel is degradable by step 1.1. By [F5], its private information is the coherent information of the average minus the average branch coherent information. Each branch term is nonnegative by [F2], so this private information is at most $I_c(\bar{\rho },{\mathcal{N}}^{\otimes n})\le n{\max }_{\rho }{I}_c(\rho ,\mathcal{N})$. Hence the $n$-use private-information supremum has the same upper bound.

Choose a maximizing state ${\rho }_{\ast }$ using [F3], and decompose it spectrally as ${\rho }_{\ast }={\sum }_j{p}_j\mathrm{\mid }{u}_j⟩⟨u_j\mathrm{\mid }$ using [F6]. For a pure input $\mathrm{\mid }{u}_j⟩⟨u_j\mathrm{\mid }$, choose a one-dimensional purification reference; the joint output-reference state has the same nonzero eigenvalues as $\mathcal{N}(\mathrm{\mid }{u}_j⟩⟨u_j\mathrm{\mid })$, so the defining formula in Complementary channels and coherent information gives $I_c(\mathrm{\mid }{u}_j⟩⟨u_j\mathrm{\mid },\mathcal{N})=0$. Therefore the pure-state ensemble has private information $I_c({\rho }_{\ast },\mathcal{N})$ by [F5]. Thus $p_1\ge {\max }_{\rho }{I}_c(\rho ,\mathcal{N})$; step 2.1 with $n=1$ gives the reverse inequality, and step 2.1 for all $n$ gives $p_n\mathrm{/}n\le {\max }_{\rho }{I}_c(\rho ,\mathcal{N})$. The regularized formula [F4] therefore yields $P(\mathcal{N})={\max }_{\rho }{I}_c(\rho ,\mathcal{N})$.

The identities in steps 1.1 and 3.1 prove the stated equality and remove both regularizations.