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Quantum iterated function systems.

Artur Łoziński1, Karol Zyczkowski, Wojciech Słomczyński

  • 1Instytut Fizyki im. Mariana Smoluchowskiego, Uniwersytet Jagielloński, ul. Reymonta 4, 30-059 Kraków, Poland. lozinski@if.uj.edu.pl

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
PubMed
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We introduce quantum iterated function systems (QIFS) for modeling nonunitary quantum dynamics. QIFS extends classical iterated function systems to Hilbert space, revealing fractal structures in quantum states.

Area of Science:

  • Quantum mechanics
  • Dynamical systems theory
  • Fractal geometry

Background:

  • Classical iterated function systems (IFS) generate fractal patterns through random function application.
  • Nonunitary quantum dynamics describes systems interacting with their environment, leading to irreversible processes.

Purpose of the Study:

  • To define and explore quantum iterated function systems (QIFS) as a framework for nonunitary quantum dynamics.
  • To investigate the properties of QIFS and their invariant states.
  • To demonstrate the emergence of fractal structures in quantum systems.

Main Methods:

  • Generalizing classical IFS to Hilbert space using completely positive maps acting on density operators.
  • Defining quantum iterated function systems (QIFS) analogous to classical IFS.

Related Experiment Videos

  • Analyzing the invariant states and measures of the defined QIFSs.

    • Main Results:

    • Exemplary classical IFSs with fractal invariant measures were presented.
    • The corresponding QIFSs and their invariant states were studied.
    • The formalism provides a method to describe nonunitary quantum dynamics.

    Conclusions:

    • Quantum iterated function systems offer a novel approach to modeling nonunitary quantum dynamics.
    • QIFS formalism can lead to fractal structures in quantum invariant states.
    • This work bridges fractal geometry and quantum dynamics.