Probability Representation of Quantum Channels

Using the known possibility to associate the completely positive maps with density matrices and recent results on expressing the density matrices with sets of classical probability distributions of dichotomic random variables we construct the probability representation of the completely positive maps. In this representation, any completely positive map of qubit state density matrix is identified with the set of classical coin probability distributions. Examples of the maps of qubit states are studied in detail. The evolution equation of quantum states is written in the form of the classical-like kinetic equation for probability distributions identified with qubit state.