 Solution

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.