发表机构
Steklov Mathematical Institute（斯捷克洛夫数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出形成纠缠的Moreau-Yosida近似函数族，证明其收敛性并给出误差上界，用于高效计算纠缠度量。
AI 中文摘要
我们描述了一族定义在双量子系统（由有限维或无限维子系统组成）状态集合上的凸一致连续函数$E^\lambda_F$，其中$\lambda>0$。当$\lambda\to0$时，这些函数单调递增并逐点收敛到形成纠缠（EoF）。这些函数是“非选择性”纠缠单调量，其构造方式类似于现代凸分析中在凸集上对凸函数进行Moreau-Yosida正则化（Moreau包络）的构建。因此，我们将函数$E^\lambda_F$称为EoF的Moreau-Yosida近似，并描述其等价定义和基本性质。EoF的半连续界（见[this http URL., 46(6), 2632-2658]）使我们能够为给定状态$\rho$下的差值$E_F(\rho)-E^\lambda_F(\rho)$获得易于计算的上界。这些界给出了函数$E^\lambda_F$在$\lambda\to0^+$时，在其中一个边缘态具有有界秩/能量的状态集合上，一致收敛到EoF的速率的易于计算的界。我们还讨论了在给定状态$\rho$下，对于所有足够小的$\lambda$，$E_F(\rho)$与$E^\lambda_F(\rho)$重合的充分条件，并考虑了几类发生这种重合的状态。简要讨论了函数$E^\lambda_F$的猜想选择性LOCC单调性及其可能的证明途径。文章的部分次要内容由ChatGPT-5.6协助撰写。
英文摘要
We describe a family of convex uniformly continuous functions $E^λ_F$, $λ>0$, on the set of states of a bipartite quantum system (consisting of finite-dimensional or infinite-dimensional subsystems), which monotonically increase and converge pointwise to the Entanglement of Formation (EoF) as $λ\to0$. These functions are "nonselective" entanglement monotones defined in a way close to the construction of the Moreau-Yosida regularization (the Moreau envelope) of a convex function on a convex set used in the modern convex analysis. So, we call the functions $E^λ_F$ the Moreau-Yosida approximations of the EoF and describe their equivalent definitions and basic properties. The semicontinuity bounds for the EoF (presented in [Lob.J.Math., 46(6), 2632-2658]) allow us to obtain easily computable upper bounds on the difference $E_F(ρ)-E^λ_F(ρ)$ for a given state $ρ$. These bounds give easily computable bounds on the rate of uniform convergence of the function $E^λ_F$ to the EoF as $λ\to0^+$ on the sets of states with bounded rank/energy of one of the marginal states. We also discuss sufficient conditions for the coincidence of $E_F(ρ)$ and $E^λ_F(ρ)$ at a given state $ρ$ for all $λ$ small enough and consider several classes of states for which such coincidence takes place. The conjectured selective LOCC-monotonicity of the functions $E^λ_F$ and a possible way to prove it are briefly discussed. Secondary parts of the article are written with the help of ChatGPT-5.6.
Comments25 pages, 2 figures, preliminary version, any comments are welcome
