量子信息理论中的Schmidt秩约束
Schmidt rank constraints in Quantum Information Theory
摘要：
具有低施密特秩的向量能形成互无偏基吗？高Schmidt秩的向量在部分转置状态下能形成正的吗？在这项工作中，我们解决这些问题，提出了几个新的结果施密特秩约束及其与其他属性的兼容性。我们提供了一个上界的相互无偏基地的$\mathbb{C}^m\otimes\mathbb{C}^n$ $(m\leq n)$形成的低施密特秩向量。特别地，$\mathbb{C}^m\otimes\mathbb{C}^n$的互无偏乘积基的个数不能超过$m+1$，解决了McNynth等人提出的一个猜想.然后，我们给出了如何从反对称空间上支持的任意态构造一个正的部分转置纠缠态，以及它们的Schmidt数是如何精确相关的.最后，我们证明了$\mathcal{M}_m\otimes \mathcal{M}_n\ (m\leq n)$在左部分转置下不变的算符Schmidt秩3态的Schmidt数不能超过$m-2$ .
英文摘要：
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 $m-2$.
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