Revisiting the operator extension of strong subadditivity

We give a new proof of the operator extension of the strong subadditivity of von Neumann entropy ${\rho }_{AB}\otimes {\sigma }_C^{-1}\le {\rho }_A\otimes {\sigma }_{BC}^{-1}$ by identifying the mathematical structure behind it as Connes’ theory of spatial derivatives. This immediately generalizes the inequality to arbitrary inclusions of von Neumann algebras. In the case of standard representations, it reduces to the monotonicity of the relative modular operator.