  Recently, Yang at al. showed that each 2-positive map acting from
$\mathcal{M}_3(\mathbb{C})$ into itself is decomposable. It is equivalent to
the statement that each PPT state on $\mathbb{C}^3\otimes\mathbb{C}^3$ has
Schmidt number at most 2. It is a generalization of Perez-Horodecki criterion
which states that each PPT state on $\mathbb{C}^2\otimes\mathbb{C}^2$ or
$\mathbb{C}^2\otimes\mathbb{C}^3$ has Schmidt rank 1 i.e. is separable. Natural
question arises whether the result of Yang at al. stays true for PPT states on
$\mathbb{C}^3\otimes\mathbb{C}^4$. We construct a positive maps which is
suspected for being a counterexample. More generally, we provide a class of
positive maps between matrix algebras whose $k$-positivity properties can be
easily controlled.
