= Solution
For a finite-dimensional memoryless <quantum channel> $\Lambda$, the <Holevo-Schumacher-Westmoreland theorem> identifies its <product-state classical capacity> with the optimized output <Holevo quantity>:
$$
\boxed{C_{\rm prod}(\Lambda)=\chi^*(\Lambda)
=\max_{\{p_x,\rho_x\}}
\left[S\left(\sum_xp_x\Lambda(\rho_x)\right)-\sum_xp_xS(\Lambda(\rho_x))\right].}
$$
With product codewords and a collective measurement on the outputs, every rate below this value is achievable with vanishing error; no larger rate can be reliable under that product-input restriction. For a fixed classical-quantum output alphabet the corresponding optimized entropy difference gives its classical coding capacity. If entangled inputs across channel uses are also allowed, the general unassisted capacity is the regularized value $C(\Lambda)=\lim_{n\to\infty}n^{-1}\chi^*(\Lambda^{\otimes n})$. The requested calculation below concerns the single-use optimized product-input expression, so no unproved additivity assumption is needed.
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