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 \(\mathbb {C}^m\otimes \mathbb {C}^n\) \((m\le n)\) 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 \(\mathcal {M}_m\otimes \mathcal {M}_n\ (m\le n)\) that are invariant under left partial transpose cannot exceed \(m-2\).

具有低施密特秩的向量可以形成相互无偏的基吗？在部分转置状态下，具有高施密特秩的向量可以形成正数吗？在这项工作中，我们通过展示与施密特秩约束及其与其他属性的兼容性相关的几个新结果来解决这些问题。我们提供了\(\mathbb {C}^m\otimes \mathbb {C}^n\) \((m\le n)\)由具有低施密特秩的向量形成的相互无偏基数的上限. 特别是\(\mathbb {C}^m\otimes \mathbb {C}^n\)的互无偏积基数不能超过\(m+1\)，它解决了 McNulty 等人提出的猜想。然后，我们展示了如何从反对称空间上支持的任何状态创建部分转置纠缠状态下的正态，以及它们的施密特数是如何精确相关的。最后，我们证明了在左部分转置下不变的\(\mathcal {M}_m\otimes \mathcal {M}_n\ (m\le n)\) 3种状态的施密特数不能超过\ (m-2\)。