= Solution
For an ensemble $\{p_x,\rho_x\}$ and measurement outcome $Y$, the <Holevo bound> is
$$
\boxed{I(X:Y)\leq\chi,
\qquad
\chi=S(\bar\rho)-\sum_xp_xS(\rho_x)},
$$
where $\bar\rho=\sum_xp_x\rho_x$. Form the <classical-quantum state> $\omega_{XB}=\sum_xp_x|x\rangle\langle x|\otimes\rho_x$. Part a gives
$$
I(X:B)_\omega=S(\bar\rho)-\sum_xp_xS(\rho_x)=\chi.
$$
The measurement is a quantum channel from $B$ to a classical register $Y$. Data processing for relative entropy, applied to the mutual-information representation, gives $I(X:Y)\leq I(X:B)$ and proves the bound.
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