论文标题
Schmidt等级信息理论中的限制
Schmidt rank constraints in Quantum Information Theory
论文作者
论文摘要
施密特等级较低的向量可以形成互无偏基的基础吗？在部分转置状态下，具有高施密特等级的载体可以形成阳性吗？在这项工作中，我们通过介绍与Schmidt等级约束及其与其他属性的兼容性有关的几个新结果来解决这些问题。我们提供了$ \ mathbb {c}^m \ otimes \ mathbb {c}^n $ $（m \ leq n）$的$ \ mathbb {c}^m \ otimes \ otimes \ otbb {c}^m \ otimes $（M \ leq n）$的上限。特别是，$ \ mathbb {c}^m \ otimes \ mathbb {c}^n $的相互无偏产品库的数量不能超过$ m+1 $，这解决了McNulty等人提出的猜想。然后，我们展示了如何从反对称空间支持的任何状态以及其Schmidt数字完全相关的任何状态下创建一个阳性状态。最后，我们证明了运算符的Schmidt数量Schmidt等级3个状态$ \ Mathcal {M} _M \ otimes \ Mathcal \ Mathcal {M} _N \（M \ LEQ N）$在左侧部分转换下是不变的。
Can vectors with low Schmidt rank form mutually unbiased bases? Can vectors with high Schmidt rank form positive under partial transpose states? In this work, we address these questions by presenting several new results related to Schmidt rank constraints and their compatibility with other properties. We provide an upper bound on the number of mutually unbiased bases of $\mathbb{C}^m\otimes\mathbb{C}^n$ $(m\leq n)$ formed by vectors with low Schmidt rank. In particular, the number of mutually unbiased product bases of $\mathbb{C}^m\otimes\mathbb{C}^n$ cannot exceed $m+1$, which solves a conjecture proposed by McNulty et al. Then we show how to create a positive under partial transpose entangled state from any state supported on the antisymmetric space and how their Schmidt numbers are exactly related. Finally, we show that the Schmidt number of operator Schmidt rank 3 states of $\mathcal{M}_m\otimes \mathcal{M}_n\ (m\leq n)$ that are invariant under left partial transpose cannot exceed $m-2$.
