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

Abstract

Whether negative-partial-transpose (NPT) states that are undistillable from one copy become distillable from finitely many copies remains a basic open problem in entanglement theory. We study the canonical two-parameter family of DiVincenzo \textit{et al.}, introduced as a symmetry-reduced testbed for this question. We prove that a distinguished one-copy-undistillable state in this family is already two-copy distillable in every local dimension $d\ge 3$. A uniform equal-norm tight-frame construction gives explicit Schmidt-rank-two certificates in every dimension, thereby disproving the conjecture that the entire one-copy-undistillable region of the canonical family remains undistillable for arbitrarily many copies. The same witnesses certify an open two-copy-distillable neighborhood around the counterexample, while separately constructed three-copy witnesses enlarge the inner bounds for the distillable region in the surrounding parameter space. In contrast, recent results for Werner states, together with the propagation argument of DiVincenzo \textit{et al.}, establish a neighboring region of one-copy-undistillable states that remains two-copy undistillable. Thus a single symmetry-reduced family contains rigorously certified states with opposite two-copy behavior, separated by a substantial region whose finite-copy distillability remains unresolved.

6+12 pages, 2 figures