Donald's identity

For an ensemble of density operators with average $\overline{\omega }={\sum }_j{p}_j{\omega }_j$,Expanding the trace definitions cancels the $\log {\omega }_j$ terms, and the remaining sum combines to a trace against $\overline{\omega }$. Positive-weight ensemble states have their support contained in that of the average, so the first sum on the right is finite in finite dimension. If the reference-state support condition fails, both sides are $+\infty$. The identity is associated with Donald's work on relative entropy.

The ensemble average $\overline{\omega }$ uniquely minimizes ${\sum }_j{p}_{j}D({\omega }_j\Vert \rho )$ over density operators $\rho$. Donald's identity separates this objective into a constant plus $D(\overline{\omega }\Vert \rho )$, and nonnegativity of quantum relative entropy proves the claim. The minimum is the Holevo quantity $S(\overline{\omega })-{\sum }_j{p}_{j}S({\omega }_j)$, equivalently the quantum mutual information of the associated classical-quantum state.