$k$-decomposability of positive maps

TL;DR

This paper introduces the concept of $k$-decomposability for positive maps between matrix algebras, providing a new framework for classifying and understanding decomposable maps, especially in low-dimensional cases.

Contribution

It proposes the novel notion of $k$-decomposability and characterizes it at the Hilbert space level, extending the classification of positive maps.

Findings

Characterization of $k$-decomposability on Hilbert spaces

Discussion of local decomposability in low-dimensional algebras

Applications for describing decomposable maps

Abstract

The problem of classification of decomposable (in the sense of Stormer) positive maps between matrix algebras is presented. We propose the new notion of "finite" version of decomposability ($k$-decomposabilty). The characterisation of $k$-decomposability on the Hilbert space level is done. In the case of low dimensional algebras the notion of local decomposability and its applications for the description of decomposable maps are discussed.