Solution

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{title2=$S(\bigoplus_ip_i\rho_i)=H(p)+\sum_ip_iS(\rho_i)$}

For <density operators> supported on mutually orthogonal subspaces, the eigenvalues of their mixture are $p_i\lambda_{ij}$, where $\lambda_{ij}$ are the eigenvalues of $\rho_i$. Substitution into the definition of <Von Neumann entropy> gives the displayed decomposition into <Shannon entropy> of the label and the mean entropy within each block. Zero eigenvalues use $0\log0=0$.

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 Incoming links: Entropy of an orthogonal quantum mixture