Improved semidefinite programming upper bound on distillable entanglement

An additive and semidefinite programming (SDP) computable entanglement measure is introduced to upper bound the amount of distillable entanglement in bipartite quantum states by operations completely preserving the positivity of partial transpose (PPT). This quantity is always smaller than or equal to the logarithmic negativity, the previously best known SDP bound on distillable entanglement, and the inequality is strict in general. Furthermore, a succinct SDP characterization of the one-copy PPT deterministic distillable entanglement for any given state is also obtained, which provides a simple but useful lower bound on the PPT distillable entanglement. Remarkably, there is a genuinely mixed state of which both bounds coincide with the distillable entanglement, while being strictly less than the logarithmic negativity.

Figure 1

This plot presents the comparison of different upper bounds on $E_{\mathrm{\Gamma }}\left({\sigma }^{\left(r\right)}\right)$ [and $E_D\left({\sigma }^{\left(r\right)}\right)$ as well]. The dashed line depicts $E_W\left({\sigma }^{\left(r\right)}\right)$ while the solid line depicts $E_N\left({\sigma }^{\left(r\right)}\right)$.