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Omit indices of probability zero. The support of a positive operator of every remaining ${\rho }_i$ lies within that of $\overline{\rho }$, because $\overline{\rho }$ is a positive sum containing $p_i{\rho }_i$. Thus $D({\rho }_i\Vert \overline{\rho })$ is finite.

First suppose $\mathrm{supp}\overline{\rho }\subseteq \mathrm{supp}\sigma$, so all quantities are finite. Expand the trace definitions of quantum relative entropy:Therefore Donald's identity isIf the support condition on $\sigma$ fails, at least one positive-weight ${\rho }_i$ also fails it. The left side and the final term on the right are then $+\infty$, while the first right-hand sum remains finite. The equality is valid in this extended sense, without taking any undefined difference of infinities.

Let $\overline{\omega }={\sum }_j{p}_j{\omega }_j$. When the relative entropies are finite, expanding their trace definitions givesThus Donald's identity isFor every $p_j>0$, the support of a positive operator ${\omega }_j$ lies in that of $\overline{\omega }$, so the first sum on the right is finite in finite dimension. If the support of $\overline{\omega }$ is not contained in the support of $\rho$, at least one positive-weight ${\omega }_j$ also violates the support condition. Both sides of the identity are then $+\infty$. This extends the conclusion without subtracting infinite quantities.

By Donald's identity, the objective equalsThe nonnegativity of quantum relative entropy makes the last term at least zero, with equality exactly when $\rho =\overline{\omega }$. Hence the unique minimizing density operator is the ensemble average andFor the classical-quantum state ${\sigma }_{XB}$, its marginal on $X$ has entropy $H(p)$, its marginal on $B$ is $\overline{\omega }$, and its joint entropy is $H(p)+{\sum }_j{p}_{j}S({\omega }_j)$. Substituting in the definition of quantum mutual information givesThis is the relative-entropy barycenter of a quantum ensemble: the average state minimizes mean distinguishability from the ensemble. Its minimal value is exactly the Holevo quantity.

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.