Nonadditivity of Rains' bound for distillable entanglement

TL;DR

This paper disproves the additivity and equality conjectures of Rains' bound for distillable entanglement, introduces a new computable lower bound $E_M$, and explores its implications for entanglement measures.

Contribution

It constructs counterexamples to Rains' bound conjectures and proposes an additive semidefinite programming lower bound for entanglement measures.

Findings

Disproved additivity of Rains' bound.

Introduced the $E_M$ lower bound for entanglement cost.

$E_M$ serves as the best known upper bound for PPT-assisted distillable entanglement.

Abstract

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.