  We address the problem of quantum process tomography with the preparators
producing states correlated with the environmental degrees of freedom that play
role in the system-environment interactions. We discuss the physical
situations, in which the dynamics is described by nonlinear, or noncompletely
positive transformations. In particular, we show that arbitrary mapping
$\varrho_{\rm in}\to\varrho_{\rm out}$ can be realized by using appropriate set
of preparators and applying the unitary operation SWAP. The experimental
``realization'' of perfect NOT operation is presented. We address the problem
of the verification of the compatibility of the preparator devices with the
estimating process. The evolution map describing the dynamics in arbitrary time
interval is known not to be completely positive, but still linear. The
tomography and general properties of these maps are discussed.
