Entropy of a classical-quantum state ($S({\sum }_x{p}_x|x⟩⟨x|\otimes {\rho }_x)=H(p)+{\sum }_x{p}_{x}S({\rho }_x)$)

The classical-quantum state is block diagonal. If ${\lambda }_{xj}$ are the eigenvalues of ${\rho }_x$, its eigenvalues are $p_x{\lambda }_{xj}$. Inserting these into the definition of Von Neumann entropy gives the displayed sum of classical Shannon entropy and mean conditional quantum entropy. Its quantum mutual information is therefore the Holevo quantity $S({\sum }_x{p}_x{\rho }_x)-{\sum }_x{p}_{x}S({\rho }_x)$.