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arXiv:quant-ph/0202153 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Irreversible quantum baker map

Artur Łoziński, Prot Pakónski, Karol Życzkowski

Published Dec 11, 2002• 17 citations
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Abstract

We propose a 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 a unitary Floquet operator. A corresponding quantum system is constructed as a completely positive map acting in the space of density matrices. We investigate spectral properties of this superoperator and their link with the increase of the entropy of initially pure states.

DOI
10.1103/physreve.66.065201

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We propose a 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 a unitary Floquet operator. A corresponding quantum system is constructed as a completely positive map acting in the space of density matrices. We investigate spectral properties of this superoperator and their link with the increase of the entropy of initially pure states.
We define a class of dynamical systems on the sphere analogous to the baker map on the torus. The classical maps are characterized by dynamical entropy equal to ln 2. We construct and investigate a family of the corresponding quantum maps. In the simplest case of the model the system does not possess a time reversal symmetry and the quantum map is represented by real, orthogonal matrices of even dimension. The semiclassical ensemble of quantum maps, obtained by averaging over a range of matrix sizes, displays statistical properties characteristic of circular unitary ensemble. Time evolution of such systems may be studied with the help of the SU (2) coherent states and the generalized Husimi distribution.
We study the behavior of an open quantum system, with an N-dimensional space of states, whose density matrix evolves according to a nonunitary map defined in two steps: A unitary step, where the system evolves with an evolution operator obtained by quantizing a classically chaotic map (baker's map and Harper's map are the two examples we consider). A nonunitary step where the evolution operator for the density matrix mimics the effect of diffusion in the semiclassical (large N) limit. The process of decoherence and the transition from quantum to classical behavior are analyzed in detail by means of numerical and analytic tools. The existence of a regime where the entropy grows with a rate that is independent of the strength of the diffusion coefficient is demonstrated. The nature of the processes that determine the production of entropy is analyzed.
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 Jamiołkowski 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.
The most general evolution of the density matrix of a quantum system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear convex set that may be viewed as supermatrices. The property of Hermiticity of density matrices renders an associated supermatrix Hermitian and hence diagonalizable. The positivity of the density matrix does not make the associated supermatrix positive though. If the map itself is positive, it is called completely positive and it has a simple parametrization. This is extended to all positive (not completely positive) maps. A general dynamical map that does not preserve the norm of the density matrices it acts on can be thought of as the contraction of a norm-preserving map of an extended system. The reconstruction of such extended dynamics is also given.
Classical chaotic systems are distinguished by their sensitive dependence on initial conditions. The absence of this property in quantum systems has lead to a number of proposals for perturbation-based characterizations of quantum chaos, including linear growth of entropy, exponential decay of fidelity, and hypersensitivity to perturbation. All of these accurately predict chaos in the classical limit, but it is not clear that they behave the same far from the classical realm. We investigate the dynamics of a family of quantizations of the baker's map, which range from a highly entangling unitary transformation to an essentially trivial shift map. Linear entropy growth and fidelity decay are exhibited by this entire family of maps, but hypersensitivity distinguishes between the simple dynamics of the trivial shift map and the more complicated dynamics of the other quantizations. This conclusion is supported by an analytical argument for short times and numerical evidence at later times.