The strong subadditivity inequality of von Neumann entropy relates the entropy of subsystems of a tripartite state ρABC to that of the composite system. Here, we define T(a)(ρABC) as the extent to which ρABC fails to satisfy the strong subadditivity inequality S(ρB)+S(ρC)≤S(ρAB)+S(ρAC) with equality and investigate its properties. In particular, by introducing auxiliary subsystem E, we consider any purification |ψABCE⟩ of ρABC and formulate T(a)(ρABC) as the extent to which the bipartite quantum correlations of ρAB and ρAC, measured by entanglement of formation and quantum discord, change under the transformation B→BE and C→CE. Invariance of quantum correlations of ρAB and ρAC under such transformation is shown to be a necessary and sufficient condition for vanishing T(a)(ρABC). 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.

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