Reexamination of strong subadditivity: A quantum-correlation approach

Abstract

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