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arXiv:2606.03561

On the saturated cases of the distillability conjecture

Saiqi Liu, Lin Chen

quant-phSubmitted Jun 2, 2026• Announced Jun 3, 2026• 0 citations
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Abstract

The distillability conjecture for two-copy four-by-four Werner states has been an open problem in quantum information for years. We investigate the conditions under which the conjectured inequality becomes an equality. For all known cases where the conjecture has been verified, we characterize the saturation conditions and show that equality forces the matrices $A$ and $B$ to be two-by-two block-diagonal. In particular, several previously obtained partial results, including the cases of one normal matrix, unitary similarity between $B$ and $-A$ or $-A^T$, and anti-diagonal block structures, are reduced to this common block-diagonal structure. We also employ a manifold optimization method, which provides numerical evidence that the two-by-two block-diagonal structure is essential for saturating the inequality. Furthermore, we prove that the identified saturation points are critical points of the objective function on the constraint manifold.

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The distillability conjecture for two-copy four-by-four Werner states has been an open problem in quantum information for years. We investigate the conditions under which the conjectured inequality becomes an equality. For all known cases where the conjecture has been verified, we characterize the saturation conditions and show that equality forces the matrices AAA and BBB to be two-by-two block-diagonal. In particular, several previously obtained partial results, including the cases of one normal matrix, unitary similarity between BBB and −A-A−A or −AT-A^T−AT, and anti-diagonal block structures, are reduced to this common block-diagonal structure. We also employ a manifold optimization method, which provides numerical evidence that the two-by-two block-diagonal structure is essential for saturating the inequality. Furthermore, we prove that the identified saturation points are critical points of the objective function on the constraint manifold.
The distillability conjecture of two-copy 4 by 4 Werner states is one of the main open problems in quantum information. We prove two special cases of the conjecture. The first case occurs when two 4 by 4 matrices A, B are both unitarily equivalent to block diagonal matrices with 2 by 2 blocks. The second case occurs when B is unitarily equivalent to either -A or the transpose of -A. Plus, we propose a simplified version of the distillability conjecture when both A and B are matrices with distinct eigenvalues.
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.
The pure entangled state is of vital importance in the field of quantum information. The process of asymptotically extracting pure entangled states from many copies of mixed states via local operations and classical communication is called entanglement distillation. The entanglement distillability problem, which is a long-standing open problem, asks whether such process exists. The 2-copy undistillability of 4×44\times44×4 undistillable Werner states has been reduced to the validness of the a matrix inequality, that is, the sum of the squares of the largest two singular values of matrix A⊗I+I⊗BA\otimes I + I \otimes BA⊗I+I⊗B does not exceed (3d−4)/d2(3d-4)/d^2(3d−4)/d2 with A,BA,BA,B traceless d×dd\times dd×d matrices and ∣∣A∣∣F2+∣∣B∣∣F2=1/d||A||_F^2+||B||_F^2=1/d∣∣A∣∣F2​+∣∣B∣∣F2​=1/d when d=4d=4d=4. The latest progress, made by Ł.~Pankowski~ et al~[IEEE Trans. Inform. Theory, 56, 4085 (2010)], shows that this conjecture holds when both matrices AAA and BBB are normal. In this paper, we prove that the conjecture holds when one of matrices AAA and BBB is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.
In this paper, we delve into a nuanced aspect of the distillability problem within the realm of entanglement theory, specifically examining the distillability of anti-diagonal [Formula: see text] entangled Werner states under the framework of local operations and classical communication (LOCC). To address this, we scrutinize a conjecture central to the distillability issue, initially focusing on its implications in 4-dimensional and 5-dimensional contexts. Through rigorous calculations, we establish that these cases align with the conjecture. Subsequently, we broaden our investigation to encompass [Formula: see text] matrices, demonstrating that both even and odd n-dimensional anti-diagonal matrices are in concordance with the conjecture, thus contributing to a deeper understanding of the distillability problem in quantum entanglement.
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.
Entanglement distillation is a fundamental task in quantum information theory. In this work, we prove that Werner states in arbitrary dimension are 222-copy distillable if and only if they are 111-copy distillable. This answers the longstanding open question of the 222-copy distillability of Werner states. This is an important step on determining whether every non-positive partial transpose (NPT) state is distillable, which remains one of the central open problems in the field of entanglement distillation.