Reexamination of strong subadditivity: A quantum-correlation approach

TL;DR

This paper introduces a quantum-correlation-based measure to analyze the failure of strong subadditivity in tripartite quantum states, linking it to quantum correlations and monogamy relations.

Contribution

It defines a new measure for the deviation from strong subadditivity using quantum correlations and characterizes states saturating the inequality through this framework.

Findings

The measure vanishes when quantum correlations are invariant under certain transformations.

States saturating strong subadditivity also saturate the Koashi-Winter monogamy relation.

Provides an intuitive characterization of states where strong subadditivity holds with equality.

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 $\boldsymbol{T}^{(a)}(\rho_{ABC})$ 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.