= Strong subadditivity from relative-entropy monotonicity
{title2=$D(\rho_{ABC}\|\rho_A\otimes\rho_{BC})\geq D(\rho_{AB}\|\rho_A\otimes\rho_B)$}
Apply <data-processing inequality for quantum relative entropy> to $\rho_{ABC}$ and $\rho_A\otimes\rho_{BC}$, tracing out $C$. The resulting comparison is $D(\rho_{ABC}\|\rho_A\otimes\rho_{BC})\geq D(\rho_{AB}\|\rho_A\otimes\rho_B)$. Expand the logarithms of the product marginals to obtain $S(AB)+S(BC)-S(B)-S(ABC)\geq0$, exactly <Strong subadditivity of Von Neumann entropy>.
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