A new operator extension of strong subadditivity of quantum entropy

Let $S(\rho )$ be the von Neumann entropy of a density matrix $\rho$. Weak monotonicity asserts that $S({\rho }_{AB})-S({\rho }_A)+S({\rho }_{BC})-S({\rho }_C)\ge 0$ for any tripartite density matrix ${\rho }_{ABC}$, 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 ${\rho }_{ABC}$, reduces to the weak monotonicity inequality. Generalizations of this inequality to the one involving two independent density matrices, as well as their Rényi-generalizations, are also presented.