基于矩的随机二分态PPT准则
Moment-based PPT criteria for random bipartite states
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中文总结 AI 辅助
研究基于矩的PPT准则在高维二分系统上的检测性能，通过研究\(\mathbb C^d\otimes\mathbb C^d\)上特定随机二分混合态，结合多种技术确定阈值环境维度，使随机态行为随维度变化，概率趋于\(1\)。
中文摘要 AI 辅助
近期引入了基于矩的正部分转置（PPT）准则松弛，作为仅涉及给定二分态实验可及量的纠缠准则层次结构。本文旨在研究其在高维二分系统上的典型检测性能。具体而言，研究了通过均匀分布纯态在环境空间\(\mathbb C^s\)上的边缘得到的\(\mathbb C^d\otimes\mathbb C^d\)上的随机二分混合态是否通常满足或违反这些准则。对于基于矩的PPT准则层次结构中的每个固定级别\(m\in\mathbb N\)，确定了一个阈值环境维度\(s = \lambda_md^2\)，随着\(d\)增大，相关随机态的行为从违反转变为满足该准则的概率趋于\(1\)。证明结合了排列组合技术估计部分转置随机态矩的平均值，以及测度集中论证来界定偏离平均值的概率，还需要通过正交多项式评估汉克尔行列式的理论工具。
英文摘要
Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on $\mathbb C^d\otimes\mathbb C^d$, obtained as the marginal over an environment $\mathbb C^s$ of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level $m\in\mathbb N$ in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension $s=λ_md^2$ at which the behavior of the associated random state switches from violating to satisfying it, with probability going to $1$ as $d$ grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension $d$ is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.
