A new operator extension of strong subadditivity of quantum entropy

Let S(?) be the von Neumann entropy of a density matrix ?. Weak monotonicity asserts that S(?(AB)) - S((?A)) + S((?BC)) - S((?C)) > 0 for any tripartite density matrix ?(ABC), a fact that is equivalent to the strong subadditivity of entropy. We prove an operator inequality, which, upon taking an expectation value with respect to the state ?(ABC), reduces to the weak monotonicity inequality. Generalizations of this inequality to the one involving two independent density matrices, as well as their Renyi-generalizations, are also presented.