We analyze a quantum analogue of an irreversible generalization of the model of classical baker map on the torus, in which the images of two parts of the phase space do overlap. This transformation is irreversible and cannot be quantized by means of an unitary Floquet operator. We construct and investigate the corresponding quantum system as a completely positive map defined by a set of measurement (Kraus) operators and acting in the space of density matrices . The quantum dynamics is non-unitary and an initially pure state suffers decoherence, which may be quantified by the von Neumann entropy of the state. We demonstrate that the initial rate of the von Neumann entropy growth depends on the KS-entropy of the classical system, provided the measurement operators have a well-defined classical limit adjusted to the classical dynamics .
 A. Lozinski, P. Pakonski and K. Zyczkowski,
"Irreversible Quantum Baker Map",
Phys. Rev. E 66, 065201(2002).
 R. Alicki, A.Lozinski, P. Pakonski and K.Zyczkowski, "Quantum dynamical entropy and decoherence rate" J. Phys.A 37, 5157(2004).