Choi representation of completely positive maps in brief

The Choi representation of completely positive (CP) maps, i.e. quantum channels is often used in the context of quantum information and computation as it is easy to work with. It is a correspondence between CP maps and quantum states also termed as the Choi-Jamiołkowski isomorphism. It is especially useful if a parametrization of the set of CP maps is needed in order to consider a general map or optimize over the set of these. Here we provide a brief introduction to this topic, focusing on certain useful calculational techniques which are presented in full detail.