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 ${\rho }_{AB}$ one has $D({\rho }_{AB}||{\sigma }_A\otimes {\sigma }_B)\ge D({\rho }_A||{\sigma }_A)+D({\rho }_B||{\sigma }_B)$. In this work, we provide an extension of this inequality for arbitrary density operators ${\sigma }_{AB}$. More specifically, we prove that $\alpha ({\sigma }_{AB})\cdot D({\rho }_{AB}||{\sigma }_{AB})\ge D({\rho }_A||{\sigma }_A)+D({\rho }_B||{\sigma }_B)$ holds for all bipartite states ${\rho }_{AB}$ and ${\sigma }_{AB}$, where $\alpha ({\sigma }_{AB})=1+2||{\sigma }_A^{-1/2}\otimes {\sigma }_B^{-1/2}{\sigma }_{AB}{\sigma }_A^{-1/2}\otimes {\sigma }_B^{-1/2}-1_{AB}|{|}_{\infty }$.

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

References in corpus (1)

• Multivariate Trace Inequalities

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