A new operator extension of strong subadditivity of quantum entropy

Abstract

Let $S(ρ)$ be the von Neumann entropy of a density matrix $ρ$. Weak monotonicity asserts that $S(ρ_{AB}) - S(ρ_A) + S(ρ_{BC}) - S(ρ_C)\geq 0$ 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 Rényi-generalizations, are also presented.