发表机构
Kyung Hee University（庆熙大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究双量子比特态谱对Schmidt数的限制，通过谱界证明最大特征值条件可保证非最大纠缠，并推广到约化判据下的态。
AI 中文摘要
Schmidt数量化了双量子比特态中纠缠的维度。我们研究了当$\nmathbb{C}^d \otimes \mathbb{C}^d$上的态谱单独能否保证其Schmidt数不是最大的。通过推导$(d-1)$-块正算子的谱界，我们证明了对于$p\leq \lfloor d^2/2\rfloor$，有$\mathrm{LS}_p\subseteq \mathrm{ASN}_{d-1}$，其中$\mathrm{LS}_p$表示最大特征值不超过其$p$个最小特征值之和的态的集合，而$\mathrm{ASN}_{d-1}$表示在所有全局酉变换下Schmidt数保持至多为$d-1$的态的集合。这也给出了一个仅涉及最大特征值的简单充分条件。作为应用，我们证明了在任意全局酉变换下满足约化判据的态在每个维度上都属于$\mathrm{ASN}_{d-1}$。这包含了先前关于绝对正部分转置态属于$\mathrm{ASN}_{d-1}$的结果。然而，全局酉变换的要求是必不可少的：对于每个$d\geq3$，我们构造了在两个子系统上都满足约化判据的最大Schmidt数的态。
英文摘要
The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on $\mathbb{C}^d \otimes \mathbb{C}^d$ alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for $(d-1)$-block positive operators, we prove that $\mathrm{LS}_p\subseteq \mathrm{ASN}_{d-1}$ for $p\leq \lfloor d^2/2\rfloor$, where $\mathrm{LS}_p$ denotes the set of states whose largest eigenvalue does not exceed the sum of their $p$ smallest eigenvalues, and $\mathrm{ASN}_{d-1}$ denotes the set of states whose Schmidt number remains at most $d-1$ under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to $\mathrm{ASN}_{d-1}$ in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to $\mathrm{ASN}_{d-1}$. The global unitary requirement is nevertheless essential: for every $d\geq3$, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.
