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 ${\rho }_{ABC}$ to that of the composite system. Here, we define ${\mathbf{T}}^{(a)}({\rho }_{ABC})$ 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 $|{\psi }_{ABCE}⟩$ of ${\rho }_{ABC}$ and formulate ${\mathbf{T}}^{(a)}({\rho }_{ABC})$ 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 $B\to BE$ 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 ${\mathbf{T}}^{(a)}({\rho }_{ABC})$. 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.

V2 (7 pages): Examples and references are added. Title is matched by the accepted version (in PRA). Original version titled "Revisit of strong ..."

References in corpus (2)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Conservation law for distributed entanglement of formation and quantum discord