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 $S(\rho_{AB}) - S(\rho_A) + S(\rho_{BC}) S(\rho_C)\geq 0$ for any tripartite $\rho_{ABC}$, fact is equivalent to strong subadditivity entropy. We prove an operator inequality, which, upon taking expectation value with respect state reduces weak inequality. Generalizations this inequality one involving two independent matrices, as well their R\'enyi-generalizations, are also presented.