支持形成纠缠（Entanglement of Formation，EoF）的仿射泛函
On supporting affine functionals for Entanglement of Formation
中文总结 AI 辅助
本文通过实例反驳了退化态下形成纠缠（EoF）存在支撑仿射泛函的假设，给出其与EoF利普希茨下半连续性的等价条件，并推导了有限秩态下EoF的利普希茨下半连续界。
中文摘要 AI 辅助
在多篇文章中，作者假设形成纠缠（EoF）的凸屋顶结构及子系统A、B的有限维性，保证了在系统AB的任意态上都存在EoF的（全局）支撑仿射泛函。这意味着，对于AB的任意态ρ，存在希尔伯特空间ℋ_AB=ℋ_A⊗ℋ_B上的厄米算子Λ_ρ，使得E_F(ρ)=Tr(Λ_ρρ)，且对AB的任意态σ，有E_F(σ)≥Tr(Λ_ρσ)。本文给出一个显式例子，表明当ρ是退化态时，即使在A、B均为量子比特系统的最简情形下，上述结论也不成立。该构造基于如下事实：在态ρ上存在EoF的支撑仿射泛函，等价于EoF在该态ρ上的利普希茨下半连续性。我们利用Wootters公式及Claude Fable 5的帮助，找到系统AB的一个态ρ，使得后者性质不成立。我们还描述了在有限维和无限维双体量子系统的给定态上，EoF的局部和全局支撑仿射泛函存在的条件。这些条件使我们能够在给定有限秩态ρ上，得到EoF的利普希茨下半连续界，即形如E_F(ρ)-E_F(σ)≤C_ρ‖ρ-σ‖₁的不等式，其中该不等式对态σ的支撑有或无限制均成立。
英文摘要
In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems $A$ and $B$ guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system $AB$. This means that for any state $ρ$ of $AB$ there is a Hermitian operator $Λ_ρ$ on $\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$ such that $E_F(ρ)=\mathrm{Tr}Λ_ρρ$ and $E_F(σ)\geq\mathrm{Tr}Λ_ρσ$ for any state $σ$ of $AB$. We present an explicit example showing that, when $ρ$ is degenerate, this is not true even in the simplest case when $A$ and $B$ are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state $ρ$ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state $ρ$. We use Wootters' formula and the help of Claude Fable 5 to find a state $ρ$ of the system $AB$ for which the latter property does not hold. We also describe conditions for the existence the local and global supporting affine functionals for the EoF at a given state of both finite and infinite-dimensional bipartite quantum systems. These conditions allow us to find Lipschitz lower semicontinuity bounds for the EoF at a given finite rank state $ρ$ (i.e. inequalities of the form $\,E_F(ρ)-E_F(σ)\leq C_ρ\|ρ-σ\|_1$) with and without restrictions on the support of the state $σ$.
