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Equality cases and every-copy results for the Werner endpoint inequality

Tejasvi Singh Tomar

Published Oct 28, 2026• 0 citations
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Abstract

Whether every entangled state with a non-positive partial transpose (NPT) is distillable reduces to the case of the Werner states ρ(d,α) = (I + αF)/(d² + αd), with F the swap, which for d ≥ 3 and −1/2 ≤ α < −1/d are NPT but one-copy undistillable. By the Schmidt-rank criterion of DiVincenzo et al. and Dür et al. (2000), as rewritten by P. Costa Rico (2025), ρ(d,α) is k-copy undistillable exactly when Σ_J α^|J| ‖Tr_J C‖² ≥ 0 for every vector of Schmidt rank at most two on k copies, written as a matrix C, where Tr_J traces out the copies in J. At α = −1/2 this inequality was proved independently for two copies by Fu et al., Fraser et al., Song and Chen, and Bharti et al. (2026), and for three copies by a computer-assisted proof in the companion record doi:10.5281/zenodo.23604343; if it held for every k, NPT bound entanglement would exist. We prove the inequality at α = −1/2 for every k and d ≥ 3 on the class of matrices C whose range is spanned by a vector f_0 of V_0 and a vector f_1 of V_1 and whose co-range (the range of C*) is arbitrary. Here V_c is spanned by |c⟩^⊗k and the k vectors obtained from it by replacing one factor with |2⟩, for orthonormal |0⟩, |1⟩, |2⟩. So no vector of this kind witnesses distillability of ρ(d,−1/2). We chose this class because, for generic f_0 and f_1, the projector onto the range has reduced states of rank three on every copy, while every known nonzero C with equality at α = −1/2 has a range with a reduced state of rank at most two on some copy. The proof reduces the claim by Schur complements to a quadratic form and then argues by induction on the number of copies. Python scripts in this record check the five rational identities of the proof symbolically; they do not involve k, and no other step uses a computer. We also prove that if the inequality holds for k − 1 copies but fails for k, then for some α > −1/2 it still holds, with equality at a nonzero C whose range has reduced states of rank at least three on every copy, as for generic members of the class. For two copies we prove that every nonzero C with equality at α = −1/2 has rank two and that the ranges of C and C* lie in products of two-dimensional single-copy subspaces. For general C the inequality remains open from four copies on.

DOI
10.5281/zenodo.23718036

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Whether every entangled state with a non-positive partial transpose (NPT) is distillable reduces to the case of the Werner states ρ(d,α) = (I + αF)/(d² + αd), with F the swap, which for d ≥ 3 and −1/2 ≤ α < −1/d are NPT but one-copy undistillable. By the Schmidt-rank criterion of DiVincenzo et al. and Dür et al. (2000), as rewritten by P. Costa Rico (2025), ρ(d,α) is k-copy undistillable exactly when Σ_J α^|J| ‖Tr_J C‖² ≥ 0 for every vector of Schmidt rank at most two on k copies, written as a matrix C, where Tr_J traces out the copies in J. At α = −1/2 this inequality was proved independently for two copies by Fu et al., Fraser et al., Song and Chen, and Bharti et al. (2026), and for three copies by a computer-assisted proof in the companion record doi:10.5281/zenodo.23604343; if it held for every k, NPT bound entanglement would exist. We prove the inequality at α = −1/2 for every k and d ≥ 3 on the class of matrices C whose range is spanned by a vector f_0 of V_0 and a vector f_1 of V_1 and whose co-range (the range of C*) is arbitrary. Here V_c is spanned by |c⟩^⊗k and the k vectors obtained from it by replacing one factor with |2⟩, for orthonormal |0⟩, |1⟩, |2⟩. So no vector of this kind witnesses distillability of ρ(d,−1/2). We chose this class because, for generic f_0 and f_1, the projector onto the range has reduced states of rank three on every copy, while every known nonzero C with equality at α = −1/2 has a range with a reduced state of rank at most two on some copy. The proof reduces the claim by Schur complements to a quadratic form and then argues by induction on the number of copies. Python scripts in this record check the five rational identities of the proof symbolically; they do not involve k, and no other step uses a computer. We also prove that if the inequality holds for k − 1 copies but fails for k, then for some α > −1/2 it still holds, with equality at a nonzero C whose range has reduced states of rank at least three on every copy, as for generic members of the class. For two copies we prove that every nonzero C with equality at α = −1/2 has rank two and that the ranges of C and C* lie in products of two-dimensional single-copy subspaces. For general C the inequality remains open from four copies on.
Motivated by the equivalent partial-trace formulations of Werner-state distillability [P. Costa Rico, Lett. Math. Phys. 115, 47 (2025); S.-Y. Qi et al., Phys. Rev. A 110, 012406 (2024)], we prove a bipartite partial-trace inequality for every matrix of rank at most two. As applications, we prove the two-copy undistillability of NPT Werner states in arbitrary local dimension, thereby resolving this open problem highlighted in [P. Horodecki et al., PRX Quantum 3, 010101 (2022)]. We further prove a two-parameter extension of the matrix inequality and show that two individually one-copy-undistillable NPT Werner states cannot activate each other's one-copy distillability. We also resolve the singular-value maximization problem associated with the two-ququart case.
A Werner state is three-copy distillable exactly when it is one-copy distillable. For every local dimension d ≥ 3, the Werner state ρ(d,α) = (I + αF)/(d² + αd) is three-copy distillable if and only if α < −1/2. The same threshold for two-copy distillability was proved in July 2026, and the qutrit case of the three-copy problem had been settled by an archived certificate. In every dimension we prove four-copy undistillability for α ≥ −79/160 and five-copy undistillability for α ≥ −9/20; for qutrits (d = 3) the bounds improve to −0.4979 and −0.47. Applied to disjoint groups of copies, the results for up to five copies give k-copy undistillability for α ≥ −11/(5k) when k ≥ 6, again in all dimensions. The endpoint α = −1/2 remains open from four copies on, and numerically the four-copy relaxation is not exact there. By a criterion of P. Costa Rico, each statement is equivalent to an inequality for complex matrices of rank at most two. We prove these inequalities with quartic moment relaxations and exact rational certificates. For three copies there is one certificate for each dimension up to 16, and one certificate whose entries are polynomials in 1/d covers every dimension d ≥ 9. For four and for five copies, one certificate each covers all dimensions from 1.25·10⁹ and from 10¹⁰ on, respectively, and zero padding passes both results to all smaller dimensions. The bound for many copies comes from a linear program over averaged partial-trace norms, with exact rational certificates. Version 3 adds results on the zeros of the endpoint form. For two copies, the range and co-range of any matrix at which the form vanishes lie in tensor products of two-dimensional subspaces, and each such plane carries a zero; in dimension four this gives all equality cases of a singular-value bound for Kronecker sums. For any number of copies the form is nonnegative near its generic zeros of this kind and, when it is nonnegative for one copy fewer, near product zeros. If k − 1 copies are settled, a failure at k copies would first appear below the endpoint, at a minimiser whose range and co-range have equal local supports of dimension at least three. A quantitative conjecture that would imply undistillability for all numbers of copies is stated and proved for two copies on one class of planes. The record contains the paper, seven verification packages with the certificates, verifiers and checks, and two Lean 4 formalizations: of the three-copy result for d = 3, …, 8, checked by Lean's kernel alone, and of the four-copy qutrit bound, in which the checks of the certificate identity are evaluated by compiled code.
We solve the two-copy distillability problem for Werner states in every local dimension. Our main matrix result is a sharp, dimension-free inequality: for every rank-at-most-two operator, the sum of the squared Hilbert--Schmidt norms of its two partial traces is bounded by twice its squared Hilbert--Schmidt norm plus one half of the squared modulus of its trace. This implies that a Werner state ραρ_αρα​ is two-copy distillable if and only if α<−1/2α<-1/2α<−1/2. In particular, the two-ququart state ρ−1/2(4)ρ^{(4)}_{-1/2}ρ−1/2(4)​ is two-copy undistillable, resolving Problem 5 of Horodecki, Rudnicki, and Życzkowski. For an arbitrary finite number kkk of copies, we give three exact formulations of the remaining problem. At the endpoint α=−1/2α=-1/2α=−1/2, undistillability is equivalent to nonnegativity of the endpoint partial-trace form on every rank-at-most-two operator. We also derive an equivalent hierarchy of operator inequalities Hk(ψ)⪰0H_k(ψ)\succeq0Hk​(ψ)⪰0 for pure states with a maximally mixed qubit marginal. The two-copy proof does not formally induct, because partial trace can increase rank and 222-positivity is not generally preserved by tensor products. We prove two rigorous many-copy extensions. First, the quadratic form factorizes exactly on tensor-factorized witnesses; for any such decomposition, the endpoint inequality holds if its possible rank-two factor is supported on a block containing at most two copies. Second, we construct explicit constants γk>0γ_k>0γk​>0 such that α≥−γkα\ge-γ_kα≥−γk​ implies kkk-copy undistillability in every dimension. The initial proofs were generated by ChatGPT 5.6 Sol; the authors have verified and rewritten them to enhance readability and provide additional context.
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≥3d\geq3d≥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.
Problem 5 in {\it Five Open Problems in Quantum Information Theory} [PRX Quantum 3, 010101 (2022)], asks whether the two-ququart Werner state ϱ(4,−12)\varrho(4,-\tfrac12)ϱ(4,−21​) is two-copy distillable, where ϱ(d,α)=(I+αF)/(d2+αd)\varrho(d,α)=(I+αF)/(d^2+αd)ϱ(d,α)=(I+αF)/(d2+αd). We answer it in the negative. To this end, we show the following stronger statement: for all C∈Md1d2(C)C\in M_{d_1d_2}(\mathbb{C})C∈Md1​d2​​(C) of rank at most r≤d1d2r \le d_1 d_2r≤d1​d2​, tr1(C)∥F2+∥tr2(C)∥F2≤r∥C∥F2+1r∣tr(C)∣2\mathrm{tr}_1(C)\|_F^2+\|\mathrm{tr}_2(C)\|_F^2 \le r\|C\|_F^2+\frac{1}{r}|\mathrm{tr}(C)|^2tr1​(C)∥F2​+∥tr2​(C)∥F2​≤r∥C∥F2​+r1​∣tr(C)∣2. A result by Costa Rico on the equivalence of this inequality with two-copy undistillability at r=2r = 2r=2 then settles Problem 5: ϱ(4,−12)\varrho(4,-\tfrac{1}{2})ϱ(4,−21​) is not two-copy distillable. Furthermore, we show that ϱ(d,α)\varrho(d,α)ϱ(d,α) is two-copy undistillable for every d≥2d\ge2d≥2, if and only if α≥−12α\ge-\tfrac{1}{2}α≥−21​. Thus, the one and two-copy distillability regions of ϱ(d,α)\varrho(d,α)ϱ(d,α) coincide. These results have been found and written up with AI tools, pointing towards a structural change affecting the field of quantum information and computation.
Whether negative-partial-transpose entanglement can remain undistillable is a longstanding problem in quantum information theory. We analyze the first unresolved three-copy Werner endpoint using a sharp dimension-free inequality for complementary partial traces and an optimal exterior-square inequality for orthonormal tripartite vectors. The latter identifies local SWAP- parity statistics with metric data of a decomposable Plucker bivector. Together these inequalities prove endpoint nonnegativity for every positive semidefinite rank-two coefficient operator and for the complete normal rank-two sector in arbitrary finite local dimensions. For genuinely nonnormal operators, an exact crossed-Gram criterion proves nonnegativity when one local outpu-input support overlap is at most two, including every system with a qubit-sized factor, and when either support plane contains a product ray. Explicit anti-state and rank-boundary families establish optimality of the constants and the rank restriction.