= Entropy of a classical-quantum state
{title2=$S(\sum_xp_x|x\rangle\langle x|\otimes\rho_x)=H(p)+\sum_xp_xS(\rho_x)$}
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> $S(\sum_xp_x\rho_x)-\sum_xp_xS(\rho_x)$.
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