AI 中文总结
本文提出Ky Fan $k$-负性作为负性的层次化推广，通过谱图优化和三次方程导出部分转置的精确谱界，并应用于认证多能级纠缠及将$p_3$-PPT条件转化为负性下界。
AI 中文摘要
在各种纠缠度量中，负性不仅因其清晰的物理意义而突出，还因其可直接从部分转置的谱计算而著称。然而，负性仅捕捉负特征值的总权重，而负谱的更精细结构在很大程度上仍未探索。在本工作中，我们通过引入负性的层次化推广——Ky Fan $k$-负性，并开发一个统一的解析框架来推导部分转置的谱界，填补了这一空白。为了获得绝对界，我们将最大化问题简化为谱图优化，其解通过单个三次方程给出每个$k$的精确界。我们进一步在固定纯度约束下，从解析和数值两方面研究这些谱界。特别地，我们完全解决了$k=1$的情况，并揭示了其下简单的图结构。作为两个直接应用，我们表明Ky Fan $k$-负性稳健地认证真正的多能级纠缠，并将$p_3$-PPT条件转化为负性的定量下界。
英文摘要
Among the various entanglement measures, the negativity stands out not only for its clear physical meaning but also for being directly computable from the spectrum of the partial transpose. However, the negativity captures only the total weight of the negative eigenvalues, whereas the finer structure of the negative spectrum remains largely unexplored. In this work, we fill this gap by introducing a hierarchical generalization of the negativity, the Ky Fan $k$-negativity, and developing a unified analytical framework for deriving spectral bounds on the partial transpose. To obtain the absolute bounds, we reduce the maximization problem to a spectral graph optimization, whose solution yields the exact bound for every $k$ through a single cubic equation. We further investigate these spectral bounds both analytically and numerically under a fixed-purity constraint. In particular, we solve the $k=1$ case completely and uncover a simple underlying graph structure. As two direct applications, we show that the Ky Fan $k$-negativity robustly certifies genuine multilevel entanglement and that it converts the $p_3$-PPT condition into a quantitative lower bound on the negativity.
Comments20 pages,5 figures
