  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 $m-2$.
