Entropy of an orthogonal quantum mixture ($S({⨁}_i{p}_i{\rho }_i)=H(p)+{\sum }_i{p}_{i}S({\rho }_i)$)

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 $0\log 0=0$.