Mapping cone of k-entanglement breaking maps

In Christandl et al. (Ann Henri Poincaré 20(7):2295–2322, 2019), 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 the completely positive maps that reduce Schmidt number on taking composition with another completely positive map. Finally, we extend a spectral majorization result for separable states.