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The relative entropy of entanglement is the smallest quantum relative entropy from a density operator to a separable quantum state, with the separable set specified for the relevant division into parties. Nonnegativity of quantum relative entropy implies $E_R\ge 0$. For a separable quantum state, choosing $\sigma =\rho$ gives $E_R=0$. For finite-dimensional bipartite states, compactness of the separable set and lower semicontinuity of the relative entropy give the converse. The support convention is $D(\rho \Vert \sigma )=+\infty$ unless $\mathrm{supp}\rho \subseteq \mathrm{supp}\sigma$.

For $0<b<1$ and distinct density operators, the Von Neumann entropy of $\rho =b{\rho }_1+(1-b){\rho }_2$ is strictly larger than the corresponding average. The difference is $bD({\rho }_1\Vert \rho )+(1-b)D({\rho }_2\Vert \rho )$, positive by nonnegativity of quantum relative entropy unless both states coincide. On the closed interval $0\le b\le 1$, equality holds exactly when $b=0$, $b=1$, or ${\rho }_1={\rho }_2$.