= Relative-entropy barycenter of a quantum ensemble
The ensemble average $\overline\omega$ uniquely minimizes $\sum_jp_jD(\omega_j\|\rho)$ over <density operators> $\rho$. <Donald's identity> separates this objective into a constant plus $D(\overline\omega\|\rho)$, and <nonnegativity of quantum relative entropy> proves the claim. The minimum is the <Holevo quantity> $S(\overline\omega)-\sum_jp_jS(\omega_j)$, equivalently the <quantum mutual information> of the associated <classical-quantum state>.
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