发表机构
Universitat Autònoma de Barcelona; Universität zu Köln; ICREA–Institució Catalana de Recerca i Estudis Avançats（巴塞罗那自治大学; 科隆大学; 加泰罗尼亚高级研究学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探究二分绝对可分态、绝对正部分转置态集合的谱几何，明确其几何性质、面与极点特征，给出纯度下界、熵的上界，证明其相对谱体积随n指数衰减。
AI 中文摘要
我们研究二分绝对可分态集合（$\text{ASEP}_{m,n}$）和绝对正部分转置态集合（$\text{APPT}_{m,n}$）的谱的几何结构，这两类态分别是在所有全局幺正变换下仍保持可分性或PPT性质的二分量子态。首先，我们建立矩阵绝对凸集、其谱及极点的一般几何性质。关于绝对可分性，我们提出绝对PPT判据的置换对称重述，并利用它证明当$m \leq n$时$\text{APPT}_{m,n}$是谱面体，其所有面均为暴露面；而$\text{ASEP}_{2,n}$也是谱面体，但一般而言，当$m \leq n$时$\text{ASEP}_{m,n}$是半代数集。此外，我们完整刻画了$\text{APPT}_{m,n}$的面和极点，证明面的维数由某一矩阵的秩决定，其最大真面的维数为$(mn - m - 1)$。在定量语境下，我们通过内接多面体$\text{mathcal{P}}_{m,n}$给出$\text{APPT}_{m,n}$可达到的最大纯度的严格下界，并猜想除$m=n=2$的情况外，任意维数下$\text{APPT}_{m,n}$的最大纯度（及其谱）均与该多面体重合。另外，我们还给出$\text{APPT}_{m,n}$最小冯·诺依曼熵的严格上界，通过数值计算证明，当局域系统维数$n$增大时，最小熵最终与多面体$\text{mathcal{P}}_{m,n}$重合。最后，我们证明$\text{APPT}_{m,n}$的相对谱体积随$n$指数衰减，衰减常数为内接多面体$\text{mathcal{P}}_{m,n}$相对体积的固定乘积因子。
英文摘要
We investigate the geometric structure of the set of spectra of bipartite absolute separable states ($\mathrm{ASEP}_{m,n}$) and absolute positive partial transpose states ($\mathrm{APPT}_{m,n}$), \emph{i.e.}, bipartite quantum states that remain separable or PPT, respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that $\mathrm{APPT}_{m,n}$ is a spectrahedron for all $m\leq n$: in particular, all its faces are exposed. While $\mathrm{ASEP}_{2,n}$ is a spectrahedron, too, for $\mathrm{ASEP}_{m,n}$ we can in general only show that it is a semialgebraic set for all $m,n\geq 3$. Furthermore, we provide a complete characterization of the faces and extreme points of $\mathrm{APPT}_{m,n}$ and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension $mn-m-1$. In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of $\mathrm{APPT}_{m,n}$ via an inscribed polytope $\mathcal{P}_{m,n}$ and conjecture that the maximal purity of $\mathrm{APPT}_{m,n}$ (along with its spectra) coincides with the polytope for arbitrary dimensions except when $m=n=2$. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of $\mathrm{APPT}_{m,n}$ and demonstrate numerically that the minimum entropy eventually coincides with the polytope $\mathcal{P}_{m,n}$ as the local system dimension $n$ increases. Finally, we show that the relative spectral volume of $\mathrm{APPT}_{m,n}$ decays exponentially in $n$ by a constant multiplicative factor of the relative volume of the inscribed polytope $\mathcal{P}_{m,n}$.
Comments48 pages, 11 figures
