= Product-input block bound for a qubit depolarizing channel
{title2=$\chi(\{p_x,\Phi^{\otimes n}(\rho_x)\})\leq n[1-h((1-p)/2)]$}
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 $n h((1-p)/2)$. 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.
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