Topology of the cone of positive maps on qubit systems

An alternative, geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra $M_{2}(\mathbb{C})$ has been presented. The premise of the proof is the identification of positive maps with operators preserving the Lorentz cone in four dimensions, and it allows to decompose the positive maps with respect to those preserving the boundary of the cone. In addition, useful conditions implying complete positivity of a map of $M_{2}(\mathbb{C})$ have been given, together with a sufficient condition for complete positivity of an extremal Schwarz map of $M_{n}(\mathbb{C})$. Lastly, following the same geometrical approach, a description in topological terms of maps that are simultaneously completely positive and completely copositive has been presented.