发表机构
Percivio Ltd.（Percivio有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文严格证明了纯化纠缠在有限张量幂下非可加，以Werner态为反例，通过解析约化与有理证书给出上下界，解决了自2002年以来的开放问题。
AI 中文摘要
纯化纠缠 $E_P$ 由 Terhal、Horodecki、Leung 和 DiVincenzo 于 2002 年提出，它通过纯化一个二分态所需的纠缠量来度量该态的总关联；自其提出以来，关于它在张量幂上是否具有可加性的问题一直悬而未决。2012 年，Chen 和 Winter 基于数值证据，以“超越合理怀疑”的力度论证了单重态分数 $f = 1/200$ 的双量子比特 Werner 态是一个反例，并明确将完全严格的证明留给了未来的工作。据我们所知，我们给出了首个严格证明：对于该态 $W$，有 $E_P^\infty(W) \le 0.9663\ldots < 97/100 < E_P(W)$，因此纯化纠缠在某个有限张量幂下是严格非可加的。下界是一个有限证书：解析约化将完整复 Stiefel 域中每个纯化的 Schmidt 谱发送到满足三维盒子中显式必要不等式的一个点，并且该盒子的精确有理细分有 25,383 个叶子，每个叶子要么排除非物理点，要么证明熵界。上界结合了一个精确的对称支撑端点、一个显式可行的等距映射，以及一个关于正则化量的凸性定理，并由两个有向有理对数证书闭合。每个计算机辅助证书都归结为整数比较；一个无依赖的程序从补充工件中验证了所有这些证书。
英文摘要
The entanglement of purification $E_P$, introduced by Terhal, Horodecki, Leung, and DiVincenzo in 2002, measures the total correlations of a bipartite state by the entanglement needed to purify it; whether it is additive on tensor powers has been open since its introduction. In 2012 Chen and Winter argued "beyond reasonable doubt", from numerical evidence, that the two-qubit Werner state at singlet fraction $f = 1/200$ is a counterexample, and explicitly left a completely rigorous proof to future work. To our knowledge we give the first rigorous proof: for this state $W$, $E_P^\infty(W) \le 0.9663\ldots < 97/100 < E_P(W)$, hence the entanglement of purification is strictly nonadditive at some finite tensor power. The lower bound is a finite certificate: analytic reductions send the Schmidt spectrum of every purification in the complete complex Stiefel domain to a point satisfying explicit necessary inequalities in a three-dimensional box, and an exact-rational subdivision of that box has 25,383 leaves, each of which either excludes nonphysical points or proves the entropy bound. The upper bound combines an exact symmetric-support endpoint, an explicit feasible isometry, and a convexity theorem for the regularized quantity, closed by two directed rational logarithm certificates. Every computer-assisted certificate reduces to integer comparisons; a dependency-free program verifies all of them from the supplementary artifact.
Comments24 pages; verification artifact archived at doi:10.5281/zenodo.22097511
