  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 $\alpha(\sigma_{AB})= 1+2 ||
\sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2}
\otimes \sigma_B^{-1/2} - \mathbb{1}_{AB} ||_\infty$.
