The three equivalent forms of completely positive maps on matrices

Abstract

Motived by the importance of quantum operations in quantum information theory, we rigorously present the three equivalent (Stinespring, Kraus, and Choi) forms of completely positive maps on full C∗-algebras of matrices, as well as their connection with the Arveson’s Radon-Nikodym derivative. In order to make this accessible to a broader audience we employ mostly linear algebra facts and carefully review the prerequisites.