Nonadditivity of Rains' bound for distillable entanglement

Rains' bound is arguably the best known upper bound of the distillable entanglement by operations completely preserving positivity of partial transpose (PPT) and was conjectured to be additive and coincide with the asymptotic relative entropy of entanglement. We disprove both conjectures by explicitly constructing a special class of mixed two-qubit states. We then introduce an additive semidefinite programming lower bound ($E_M$) for the asymptotic Rains' bound, and it immediately becomes a computable lower bound for entanglement cost of bipartite states. Furthermore, $E_M$ is also proved to be the best known upper bound of the PPT-assisted deterministic distillable entanglement and gives the asymptotic rates for all pure states and some class of genuinely mixed states.

Figure 1

This plot demonstrates the difference between $2R\left({\rho }_r\right)$ and $E_R^{+}\left({\rho }_r^{\otimes 2}\right)$ for $0.45\le r\le 0.548$. The dashed line depicts $E_R^{+}\left({\rho }_r^{\otimes 2}\right)$, while the solid line depicts $2R\left({\rho }_r\right)$.

Figure 2

This plot presents the estimation of $E_{D,\mathrm{PPT}}\left({\rho }^{\left(\alpha \right)}\right)$ and $E_{0,D,\mathrm{PPT}}\left({\rho }^{\left(\alpha \right)}\right)$. The dotted line depicts $E_W\left({\rho }^{\left(\alpha \right)}\right)$, the dashed line depicts $E_{0,D,\mathrm{PPT}}^{\left(1\right)}\left({\rho }^{\left(\alpha \right)}\right)$, and the solid line depicts $E_M\left({\rho }^{\left(\alpha \right)}\right)$.