  The simplest building blocks for quantum computations are the qbit-qbit
quantum channels. In this paper we analyse the structure of these channels via
their Choi representation. The restriction of a quantum channel to the space of
classical states (i.e. probability distributions) is called the underlying
classical channel. The structure of quantum channels over a fixed classical
channel is studied, the volume of general and unital qubit channels over real
and complex state spaces with respect to the Lebesgue measure is computed and
explicit formulas are presented for the distribution of the volume of quantum
channels over given classical channels. Moreover an algorithm is presented to
generate uniformly distributed channels with respect to the Lebesgue measure,
which enables further studies. With this algorithm the distribution of
trace-distance contraction coefficient (Dobrushin) is investigated numerically
by Monte-Carlo simulations, which leads to some conjectures and points out the
strange behaviour of the real state space.
