Mapping cone of $k$-Entanglement Breaking Maps

TL;DR

This paper systematically studies $k$-entanglement breaking maps, providing equivalent conditions, analyzing their cone structure, and characterizing maps that reduce Schmidt number upon composition.

Contribution

It offers a comprehensive analysis of $k$-entanglement breaking maps, including equivalent conditions and their cone structure, advancing understanding of entanglement properties.

Findings

Established multiple equivalent conditions for $k$-entanglement breaking maps

Analyzed the mapping cone structure of these maps

Characterized maps reducing Schmidt number upon composition

Abstract

In \cite{CMW19}, the authors introduced $k$-entanglement breaking linear maps to understand the entanglement breaking property of completely positive maps on taking composition. In this article, we do a systematic study of $k$-entanglement breaking maps. We prove many equivalent conditions for a $k$-positive linear map to be $k$-entanglement breaking, thereby study the mapping cone structure of $k$-entanglement breaking maps. We discuss examples of $k$-entanglement breaking maps and some of their significance. As an application of our study, we characterize completely positive maps that reduce Schmidt number on taking composition with another completely positive map.