 Solution

Let ${\gamma }_1$ and ${\gamma }_2$ be the individual Ryu–Takayanagi surfaces for $D_1$ and $D_2$. Their disconnected union ${\gamma }_1\cup {\gamma }_2$ is homologous to $D_1\cup D_2$ and is therefore an admissible competitor in the minimization that defines ${\gamma }_{12}$. Minimality givesDividing by $4G$ provesand hence the leading holographic mutual information obeysThis is the geometric realization of the nonnegativity of quantum mutual information.