On a class of $k$-entanglement witnesses

Marcin Marciniak, Tomasz Młynik, Hiroyuki Osaka

Published 2021-04-29Version 1

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.