 Product-input block bound for a qubit depolarizing channel

For $\Phi (\rho )=p\rho +(1-p)I/2$, every output qubit has Von Neumann entropy at least $h((1-p)/2)$. A product input therefore has output entropy at least $nh((1-p)/2)$. The average output has entropy at most $n$, so its Holevo quantity is bounded by the displayed expression. This allows correlated classical message labels and joint output measurements, but requires product quantum inputs. The Holevo bound and Fano's inequality turn it into the corresponding asymptotic rate bound.