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The Umegaki relative entropy iswhen the support of $\rho$ is contained in that of $\sigma$, and $+\infty$ otherwise. Its additivity of quantum relative entropy isits superadditivity of quantum relative entropy isand its data-processing inequality for quantum relative entropy is $D(T(\rho )\Vert T(\sigma ))\le D(\rho \Vert \sigma )$ for every quantum channel $T$.

Additivity follows from the logarithm of a tensor product,and the analogous identity for ${\sigma }_A\otimes {\sigma }_B$. For superadditivity, subtract the two marginal relative entropies from the joint one. The reference-state terms cancel, leavingThis is the nonnegativity of quantum mutual information, equivalently Subadditivity of Von Neumann entropy.

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.

For a bipartite density operator ${\rho }_{AB}$ and a product reference state,The difference between the two sides is the nonnegative quantum mutual information of ${\rho }_{AB}$.