# Superadditivity of quantum relative entropy for general states

## TL;DR

This paper extends the superadditivity property of quantum relative entropy to arbitrary states, introducing a new inequality involving a state-dependent factor that bounds the sum of marginal divergences.

## Contribution

It generalizes the superadditivity inequality for quantum relative entropy to all density operators, incorporating a state-dependent coefficient for broader applicability.

## Key findings

- Established a new inequality for quantum relative entropy with arbitrary states.
- Derived a state-dependent coefficient that bounds the superadditivity relation.
- Provides theoretical foundation for analyzing quantum correlations in 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 $\alpha(\sigma_{AB})= 1+2 || \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} - \mathbb{1}_{AB} ||_\infty$.