Two-copy distillability of one-copy-undistillable negative-partial-transpose states in every dimension

Abstract

Whether entanglement with a negative partial transpose (NPT) can be undistillable is a longstanding open problem. In the canonical family of DiVincenzo \textit{et al.}, the one-copy-undistillable region was conjectured to remain undistillable at all copy numbers. We refute this: in every dimension $d\ge 3$, one vertex of this region is two-copy distillable, via an explicit Schmidt-rank-two tight-frame witness. The separation persists on an open set, while other states in the same region are provably two-copy undistillable. Twirling our counterexample into Werner form makes it two-copy undistillable, so this standard reduction can erase finite-copy distillability.

6+16 pages, 2 figures. Added exact rational two-copy-undistillability certificates for regions inside $BCG$ for $d=3,\dots ,8$, plus further finite-copy and numerical analysis