Reexamination of strong subadditivity: A quantum-correlation approach

The strong subadditivity inequality of von Neumann entropy relates the entropy of subsystems of a tripartite state ${\rho }_{ABC}$ to that of the composite system. Here, we define ${\mathit{T}}^{\left(a\right)}\left({\rho }_{ABC}\right)$ as the extent to which ${\rho }_{ABC}$ fails to satisfy the strong subadditivity inequality $S\left({\rho }_B\right)+S\left({\rho }_C\right)\le S\left({\rho }_{AB}\right)+S\left({\rho }_{AC}\right)$ with equality and investigate its properties. In particular, by introducing auxiliary subsystem $E$, we consider any purification $|{\psi }_{ABCE}\rangle$ of ${\rho }_{ABC}$ and formulate ${\mathit{T}}^{\left(a\right)}\left({\rho }_{ABC}\right)$ as the extent to which the bipartite quantum correlations of ${\rho }_{AB}$ and ${\rho }_{AC}$, measured by entanglement of formation and quantum discord, change under the transformation $B\to BE$ and $C\to CE$. Invariance of quantum correlations of ${\rho }_{AB}$ and ${\rho }_{AC}$ under such transformation is shown to be a necessary and sufficient condition for vanishing ${\mathit{T}}^{\left(a\right)}\left({\rho }_{ABC}\right)$. Our approach allows one to characterize, intuitively, the structure of states for which the strong subadditivity is saturated. Moreover, along with providing a conservation law for quantum correlations of states for which the strong subadditivity inequality is satisfied with equality, we find that such states coincide with those that the Koashi-Winter monogamy relation is saturated.

Figure 1

(a) ${\mathit{T}}^{\left(a\right)}\left({\rho }_{ABC}\right)$ in terms of ${\beta }_2$ and ${\lambda }_1$, for $b=1/\sqrt{2}$. (b) ${\mathit{T}}^{\left(a\right)}\left({\rho }_{ABC}\right)$ in terms of ${\beta }_2$ and $b$, for ${\lambda }_1=1/2$. In both plots, we assumed ${\alpha }_1=1/\sqrt{2}$ and $p_1={\lambda }_2=1/2$.