  We investigate the space of quantum operations, as well as the larger space
of maps which are positive, but not completely positive. A constructive
criterion for decomposability is presented. A certain class of unistochastic
operations, determined by unitary matrices of extended dimensionality, is
defined and analyzed. Using the concept of the dynamical matrix and the
Jamiolkowski isomorphism we explore the relation between the set of quantum
operations (dynamics) and the set of density matrices acting on an extended
Hilbert space (kinematics). An analogous relation is established between the
classical maps and an extended space of the discrete probability distributions.
