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arXiv:quant-ph/0210029 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum iterated function systems

Artur Łoziński, Karol Życzkowski, Wojciech Słomczyński

Published Oct 13, 2003• 36 citations
arXivHTMLPDF

Abstract

An iterated function system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum IFS (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting, a QIFS consists of completely positive maps acting in the space of density operators. This formalism is designed to describe certain problems of nonunitary quantum dynamics. We present exemplary classical IFSs, the invariant measure of which exhibits fractal structure, and study properties of the corresponding QIFSs and their invariant states.

DOI
10.1103/physreve.68.046110

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An iterated function system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum IFS (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting, a QIFS consists of completely positive maps acting in the space of density operators. This formalism is designed to describe certain problems of nonunitary quantum dynamics. We present exemplary classical IFSs, the invariant measure of which exhibits fractal structure, and study properties of the corresponding QIFSs and their invariant states.
Iterated Function Systems (IFS) generate invariant measures with fractal structures through the random iteration of a family of contraction mappings, servingas a standard framework for describing scale generation phenomena in classicalnonlinear dynamics. This paper generalizes classical IFS to the quantum domain,establishing a rigorous mathematical formulation of Quantum Iterated FunctionSystems (QIFS): classical contraction mappings are replaced by families of completely positive maps acting on the space of density operators, and the quantumattractor state is defined as the fixed point satisfying a self-similarity equation. Onthis basis, we prove that the rescaled position distribution of discrete-time quantum random walks with decoherence converges to a convex combination of normaldistributions, with the convergence rate controlled by the spectral gap of the superoperator. Furthermore, we establish a quantum-classical correspondence principle:in the limit of vanishing decoherence rate, the support of the Wigner function ofthe quantum attractor state converges in distribution to the classical IFS attractor,and the convergence exponents satisfy a scaling relation with the classical contraction rates. This framework unifies fractal geometry, quantum open systems, anddecoherence theory into a coherent generative language, providing a rigorous mathematical foundation for understanding scaling behavior in quantum informationprocessing.
The mathematical structure of quantum groups, as q-deformations of Lie algebras, has been extensively studied, yet their deep connection with the dynamicsof open quantum systems remains to be elucidated. Based on the framework ofQuantum Iterated Function Systems (QIFS), this paper establishes an emergencemechanism from quantum groups to classical Lie groups. We first construct a concrete QIFS whose generators are a family of completely positive maps acting onthe space of quantum states, and prove that the Lie algebra structure of its lowenergy effective theory corresponds to the Drinfeld–Jimbo quantum group Uq(g).Next, we derive the relation between the q-parameter and the decoherence rate Γ,characteristic coherence time τc as q = e−Γτc, revealing the physical origin of theq-deformation. Based on this, we prove that in the limit of vanishing decoherencerate Γ → 0, the quantum group degenerates to the classical Lie group g. Finally, weestablish a correspondence between the representation theory of quantum groupsand the spectrum of QIFS, providing a unified mathematical framework for understanding the emergence mechanism of quantum symmetries. This result links theconcept of decoherence in quantum information theory with the algebraic structureof quantum groups, revealing a deeper picture where classical Lie group symmetriesemerge as the semiclassical limit of open quantum system dynamics.
Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of com-plex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geomet-rically induced conformal maps (Möbius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by “density matrices ” in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two–sphere (octahedron), and on three-dimensional sphere (hypercube-tesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we discuss the Hamilton’s “icossian calculus ” and its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. 1 1
An intriguing and deep analogy between classical and quantum states is revealed using the notion of positive definite function. By Bochner's theorem, (classical) positive definite functions on phase space can be obtained by taking the Fourier transform of probability measures; similarly, quantum positive definite functions are the image via the Fourier-Plancherel operator of Wigner quasi-probability distributions. Considering the basic properties of positive definite functions — classical and quantum — one is led to define a class of semigroups of operators, the so-called classical-quantum semigroups. It is then natural to wonder whether they have any physical meaning. It turns out that the classical-quantum semigroups can also be obtained by dequantizing a certain class of quantum dynamical semigroups, namely, the classical noise semigroups. This correspondence fits in a more general group-theoretical framework in which a larger class of quantum dynamical semigroups, the twirling semigroups, can be suitably dequantized. Connections with quantum information science will be briefly discussed.