基于迹不变量的最优纠缠判据
Optimal entanglement criteria from trace invariants
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中文总结 AI 辅助
本文通过凸对偶方法，基于迹不变量构建并分析了最优纠缠判据，确定了可分锥的极射线结构，并提出了新的可分性不等式层级，为纠缠检测提供了理论框架。
中文摘要 AI 辅助
随机测量协议能够实验上获取量子态的低次多项式不变量，而非量子态本身，这引发了一个问题：由有限个这样的不变量能够构建的最优纠缠判据是什么。在本工作中，我们通过凸对偶回答了这一问题。对于每个次数，我们引入了对所有局域维度下可分两体态均有效的局域酉迹不等式锥，以及其对于厄米矩阵和半正定矩阵的单矩阵对应锥。每个锥都是某个矩锥的极对偶；我们还确定了可分锥的最小组合参数化。该框架整理了已知的矩判据：利用部分转置的前三阶矩的PPT判据，是半正定锥到两体锥的Hankel行列式在嵌入下的像，我们证明了该像是一条极射线，并完全确定了第一个非平凡可分锥，发现恰好有四条极射线。该框架使我们能够分析重排判据及其中心化（或增强）变体。我们证明了（中心化）重排矩阵的偶数奇异值矩是迹不变量，与显式的无不动点对合相关联。因此，从$m$个这样的矩中提取最优迹范数界成为一个无维数的截断矩问题，我们对于$m=2$给出了闭式解，并对任意$m$松弛为规模为$O(m)$的半定规划。对偶解是$[0,1]$上$\sqrt{x}$的多项式下界函数，两个显式族——二项式族和$L^1$加权族——产生了可分性不等式层级。最后，我们将这些判据相互比较，并应用于PPT纠缠态。
英文摘要
Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality. For each degree we introduce the cone of local unitary trace inequalities that are valid for separable bipartite states in all local dimensions, together with its one-matrix counterparts for Hermitian and positive semidefinite matrices. Each cone is the polar dual of a moment cone; we also determine the minimal combinatorial parametrization of the separable cone. The framework organizes the known moment criteria: the PPT criterion using the first three moments of the partial transposition is the image under an embedding of a Hankel determinant from the positive semidefinite cone to the bipartite cone, which we show to be an extremal ray, and we determine the first nontrivial separable cone completely, finding exactly four extremal rays. The framework allows us to analyze the realignment criterion and its centered (or enhanced) variant. We prove that the even singular value moments of the (centered) realignment matrix are trace invariants, associated with explicit fixed-point-free involutions. Extracting the optimal trace-norm bound from $m$ such moments thus becomes a dimension-free truncated moment problem, which we solve in closed form for $m = 2$ and relax to a semidefinite program of size $O(m)$ for arbitrary $m$. The dual solutions are polynomial minorants of $\sqrt{x}$ on $[0,1]$, and two explicit families, binomial and $L^1$-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.
发表机构
- S. N. Bose National Centre for Basic Sciences（S.N. 玻色基础科学国家中心）
- Indian Statistical Institute（印度统计研究所）
- Université de Toulouse（图卢兹大学）
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS（理论物理实验室，图卢兹大学，法国国家科学研究中心）
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