Undistillability of Werner states from three, four and five copies

Whether all entangled states with a non-positive partial transpose (NPT) can be distilled is open. The question reduces to the Werner states ρ(d,α) = (I + αF)/(d² + αd), where F is the swap. For d ≥ 3 and −1/2 ≤ α < −1/d these states are NPT but one-copy undistillable. If ρ(d,−1/2) were undistillable from any number of copies, NPT bound entanglement would exist. That two copies do not suffice on this range was proved independently by Fu et al. (arXiv:2607.21367), Fraser et al. (arXiv:2607.24309), Song and Chen (arXiv:2607.23416) and Bharti et al. (arXiv:2607.24479).

Our main result is that for all d ≥ 3, ρ(d,α) is three-copy distillable if and only if α < −1/2, so three copies do no better than one. This contains the two-copy result, which our three-copy proof does not use. In all dimensions we prove four-copy undistillability for α ≥ −79/160 and five-copy undistillability for α ≥ −9/20. For qutrits we improve these bounds to −4979/10000 and −47/100. With k ≥ 6 copies we prove undistillability for α ≥ −11/(5k). The previous best bounds, by Bharti et al., have the form α ≥ −γ_k with kγ_k → log φ ≈ 0.481, so the constant 11/5 is about 4.6 times larger, and our range contains NPT states when k < 11d/5. We do not know whether our bounds for four or more copies are sharp.

By the Schmidt-rank criterion of DiVincenzo et al. and Dür et al. (Phys. Rev. A 61, 062312 and 062313, 2000), in the form given by P. Costa Rico (Lett. Math. Phys. 115, 47, 2025), k-copy undistillability means that for each vector of Schmidt rank at most two, written as a matrix, an alternating sum of the squared norms of its partial traces is nonnegative. We prove these inequalities with exact certificates for fourth-moment semidefinite relaxations whose data are polynomial in 1/d. For three copies one certificate covers all d ≥ 9. For four and five copies one certificate of the limiting relaxation as d → ∞ covers all sufficiently large d, since its error at finite d is bounded exactly. Since the inequality in one dimension implies it in each smaller one, this settles all d ≥ 3. For qutrits, exact certificates at d = 3 give the sharper four- and five-copy bounds. For k ≥ 6 copies we apply the inequalities for up to five copies to groups of copies and combine them in a linear program with known partial-trace inequalities, among them the two-copy one.

The record contains the paper, its verification packages, two Lean 4 formalizations and a supplement. The proofs rely on the packages: Python programs that check each certificate in exact or ball arithmetic, mostly with python-flint. Their READMEs give the commands. As an independent check, the Lean formalizations re-prove the four-copy qutrit bound and the three-copy inequality for d = 3, …, 8, which the paper obtains from d = 9. The supplement describes these checks and adds independent ones, such as exact three-copy certificates at fixed dimensions. A companion record (doi:10.5281/zenodo.23718037) proves the inequality at α = −1/2 for all k on a class of matrices whose ranges are spanned by one vector from each of two explicit (k+1)-dimensional subspaces.