Abstract
Let S(ρ) be the von Neumann entropy of a density matrix ρ. Weak monotonicity asserts that S(ρ_AB) - S(ρ_A) + S(ρ_BC) - S(ρ_C)≥ 0 for any tripartite density matrix ρ_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 ρ_ABC, reduces to the weak monotonicity inequality. Generalizations of this inequality to the one involving two independent density matrices, as well as their R\'enyi-generalizations, are also presented.