Superadditivity of quantum relative entropy for general states

Abstract

The property of superadditivity of the quantum relative entropy states that, in a bipartite system $\mathcal{H}_{AB}=\mathcal{H}_A \otimes \mathcal{H}_B$, for every density operator $ρ_{AB}$ one has $ D( ρ_{AB} || σ_A \otimes σ_B ) \ge D( ρ_A || σ_A ) +D( ρ_B || σ_B) $. In this work, we provide an extension of this inequality for arbitrary density operators $ σ_{AB} $. More specifically, we prove that $ α(σ_{AB})\cdot D({ρ_{AB}}||{σ_{AB}}) \ge D({ρ_A}||{σ_A})+D({ρ_B}||{σ_B})$ holds for all bipartite states $ρ_{AB}$ and $σ_{AB}$, where $α(σ_{AB})= 1+2 || σ_A^{-1/2} \otimes σ_B^{-1/2} \, σ_{AB} \, σ_A^{-1/2} \otimes σ_B^{-1/2} - \mathbb{1}_{AB} ||_\infty$.

14 pages. v3: Final version. The main theorem has been improved, adding a fourth step to its proof and also some remarks. v2: There was a flaw in the proof of the previous version. This has been corrected in this version. The constant appearing in the main Theorem has changed accordingly