Quantization of irreversible chaotic maps

Karol Zyczkowski

Perimeter Institute

and

Institute of Physics, Jagiellonian University, Cracow, Poland

Abstract:

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 [1].
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
[2].

[1] A. Lozinski, P. Pakonski and K. Zyczkowski,
"Irreversible Quantum Baker Map",
Phys. Rev. E 66, 065201(2002).

[2] R. Alicki, A.Lozinski, P. Pakonski and K.Zyczkowski,
"Quantum dynamical entropy and decoherence rate"
J. Phys.A 37, 5157(2004).