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 $\boldsymbol{T}^{(a)}(\rho_{ABC})$ as the extent to which $\rho_{ABC}$ fails to satisfy the strong subadditivity inequality $S(\rho_{B})+S(\rho_{C}) \le S(\rho_{AB})+S(\rho_{AC})$ 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 $\boldsymbol{T}^{(a)}(\rho_{ABC})$ 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 $B\rightarrow BE$ and $C\rightarrow CE$. Invariance of quantum correlations of $\rho_{AB}$ and $\rho_{AC}$ under such transformation is shown to be a necessary and sufficient condition for vanishing $\boldsymbol{T}^{(a)}(\rho_{ABC})$. 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 coincides with those that the Koashi-Winter monogamy relation is saturated.