There is no direct generalization of positive partial transpose criterion to the three-by-three case

Abstract

We show that there cannot exist a straightforward generalization of the famous positive partial transpose criterion to three-by-three systems. We call straightforward generalizations that use a finite set of positive maps and arbitrary local rotations of the tested two-partite state. In particular, we show that a family of extreme positive maps discussed in a paper by Ha and Kye [Open Syst. Inf. Dyn. 18, 323–337 (2011)], cannot be replaced by a finite set of witnesses in the task of entanglement detection in three-by-three systems. In a more mathematically elegant parlance, our result says that the convex cone of positive maps of the set of three-dimensional matrices into itself is not finitely generated as a mapping cone.