= Solution
The <Holevo-Schumacher-Westmoreland theorem> gives the classical coding formula
$$
C(\Phi)=\lim_{m\to\infty}\frac1m\chi^*(\Phi^{\otimes m}),\qquad\chi^*(\Lambda)=\sup_{\{p_i,\rho_i\}}\chi(\{p_i,\Lambda(\rho_i)\}).
$$
Its ensemble-coding statement makes every rate below a one-use output <Holevo quantity> achievable with product input encodings and collective output decoding. Under this product-input restriction, the capacity is
$$
C_{\mathrm{prod}}(\Phi)=\sup_{\{p_i,\rho_i\}}\chi(\{p_i,\Phi(\rho_i)\}).
$$
For any such input ensemble, rates below its output <Holevo quantity> are achievable with asymptotically vanishing decoding error. Maximization gives the <product-state classical capacity>. The converse follows from the <Holevo bound>, entropy bounds on product outputs, and <Fano's inequality>.
Define the <binary entropy> $h_2(x)=-x\log_2x-(1-x)\log_2(1-x)$. For an input Bloch radius $r\leq1$, the output eigenvalues are $(1\pm(1-q)r)/2$. Its entropy is
$$
S(\Phi(\rho))=h_2\left(\frac{1+|1-q|r}{2}\right).
$$
On $[1/2,1]$, $h_2$ is decreasing, so this is minimized by pure inputs $r=1$. Throughout the completely positive interval $0\leq q\leq4/3$, the minimum equals $h_2(q/2)$, using symmetry $h_2(x)=h_2(1-x)$. The average output entropy is at most $1$, because it is a qubit. Hence every ensemble satisfies
$$
\chi(\{p_i,\Phi(\rho_i)\})\leq1-h_2(q/2).
$$
Two equally likely orthogonal pure inputs attain this bound: their average output is $I/2$, and each individual output has entropy $h_2(q/2)$. Therefore
$$
\boxed{C_{\mathrm{prod}}(\Phi)=1-h_2(q/2)=1-h_2(2p/3)\quad\text{bits per channel use}.}
$$
This gives the <Holevo capacity of a qubit depolarizing channel> in the noise-weight convention used here.
For completeness, correlated classical code labels do not evade this upper bound. Each product input to $n$ uses has product output entropy at least $nh_2(q/2)$, whereas the average output entropy is at most $n$. Its <Holevo quantity> is consequently at most $n[1-h_2(q/2)]$. The <Holevo bound> and <Fano's inequality> rule out a reliable rate larger than this value. Combined with HSW achievability, this proves the capacity under the stated product-input restriction. It also remains valid in the extended physical parameter range arising from the original $p$ assumption.
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