Random and free positive maps with applications to entanglement detection

TL;DR

This paper introduces new families of $k$-positive maps using random matrix and free probability techniques, enhancing entanglement detection methods beyond partial transposition, especially for PPT entangled states.

Contribution

It develops parametrized families of $k$-positive maps, linking random matrix theory with free probability, and demonstrates their effectiveness in detecting PPT entanglement.

Findings

New $k$-positive maps are constructed and characterized.

Some maps are indecomposable and can detect PPT entanglement.

Refined understanding of PPT states in large-small dimensional settings.

Abstract

We apply random matrix and free probability techniques to the study of linear maps of interest in quantum information theory. Random quantum channels have already been widely investigated with spectacular success. Here, we are interested in more general maps, asking only for $k$-positivity instead of the complete positivity required of quantum channels. Unlike the theory of completely positive maps, the theory of $k$-positive maps is far from being completely understood, and our techniques give many new parametrized families of such maps. We also establish a conceptual link with free probability theory, and show that our constructions can be obtained to some extent without random techniques in the setup of free products of von Neumann algebras. Finally, we study the properties of our examples and show that for some parameters, they are indecomposable. In particular, they can be used to detect the presence of entanglement missed by the partial transposition test, that is, PPT entanglement. As an application, we considerably refine our understanding of PPT states in the case where one of the spaces is large whereas the other one remains small.