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For an ensemble $\{p_x,{\rho }_x\}$ with average $\bar{\rho }$, the Holevo quantity is $\chi =S(\bar{\rho })-{\sum }_x{p}_{x}S({\rho }_x)=I(X:B)$ of the associated classical-quantum state.

Both states are block diagonal in $X$. On the support of each block,Substitution into quantum relative entropy givesTaking $-\mathrm{Tr}{\rho }_{XB}\log {\rho }_{XB}$ similarly yields the entropy of a classical-quantum state:

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.