= Entropy bounds for a quantum mixture
{title2=$\sum_ip_iS(\rho_i)\leq S(\sum_ip_i\rho_i)\leq H(p)+\sum_ip_iS(\rho_i)$}
= Entropy bound for a quantum mixture
{synonym}
For a mixture of <density operators>, the lower bound follows from $S(\bar\rho)-\sum_ip_iS(\rho_i)=\sum_ip_iD(\rho_i\|\bar\rho)\geq0$, with equality precisely when all positive-weight states coincide. For the upper bound, resolve each state into pure eigenvectors and form the Gram matrix of the resulting weighted vectors. Its nonzero eigenvalues are those of $\bar\rho$, while its diagonal has probabilities $p_i\lambda_{ia}$. Averaging diagonal unitary conjugations deletes its off-diagonal entries; <Concavity of Von Neumann entropy> therefore gives the bound. Equality holds exactly when positive-weight component supports are mutually orthogonal.
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