 Solution

For an ensemble $\{p_x,{\rho }_x\}$ and measurement outcome $Y$, the Holevo bound iswhere $\bar{\rho }={\sum }_x{p}_x{\rho }_x$. Form the classical-quantum state ${\omega }_{XB}={\sum }_x{p}_x|x⟩⟨x|\otimes {\rho }_x$. Part a givesThe measurement is a quantum channel from $B$ to a classical register $Y$. Data processing for relative entropy, applied to the mutual-information representation, gives $I(X:Y)\le I(X:B)$ and proves the bound.