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

TL;DR

This paper proves that the positive partial transpose criterion cannot be straightforwardly extended to three-by-three quantum systems, highlighting fundamental limitations in entanglement detection methods for higher-dimensional states.

Contribution

It demonstrates that no finite set of positive maps can generalize the PPT criterion for three-by-three systems, revealing intrinsic mathematical constraints.

Findings

No finite set of positive maps suffices for entanglement detection in 3x3 systems.

The convex cone of positive maps in 3D matrices is not finitely generated.

Straightforward generalizations of PPT do not exist for higher dimensions.

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, 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