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 gives
$$
\operatorname{Area}(\gamma_{12})
\leq\operatorname{Area}(\gamma_1)
+\operatorname{Area}(\gamma_2).
$$
Dividing by $4G$ proves
$$
S(D_1\cup D_2)\leq S(D_1)+S(D_2),
$$
and hence the leading <holographic mutual information> obeys
$$
\boxed{I_c\geq0}.
$$
This is the geometric realization of the nonnegativity of <quantum mutual information>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 2 / b / i / Solution

Holographic mutual information is nonnegative because the disconnected union $\gamma_A\cup\gamma_B$ is an admissible surface for $A\cup B$, whose minimizing surface can have no larger area.


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 Incoming links: Holographic mutual information