  One of the most challenging open problems in quantum information theory is to
clarify and quantify how entanglement behaves when part of an entangled state
is sent through a quantum channel. Of central importance in the description of
a quantum channel or completely positive map (CP-map) is the dual state
associated to it. The present paper is a collection of well-known, less known
and new results on quantum channels, presented in a unified way. We will show
how this dual state induces nice characterizations of the extremal maps of the
convex set of CP-maps, and how normal forms for states defined on a Hilbert
space with a tensor product structure lead to interesting parameterizations of
quantum channels.
