Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access September 1, 2004

On quantum iterated function systems

Arkadiusz Jadczyk
From the journal Open Physics

Abstract

A Quantum Iterated Function System on a complex projective space is defined through a family of linear operators on a complex Hilbert space. The operators define both the maps and their probabilities by one algebraic formula. Examples with conformal maps (relativistic boosts) on the Bloch sphere are discussed.

[1] M.F. Barnsley: Fractals everywhere, Academic Press, San Diego, 1988. Search in Google Scholar

[2] L. Skala, K. Bradler and V. Kapsa: “Consistency requirement and operators in quantum mechanics”, Czech. J. Phys., Vol.52, (2002), pp.345–350. http://dx.doi.org/10.1023/A:101452391721210.1023/A:1014523917212Search in Google Scholar

[3] A. Jadczyk and R. Öberg: “Quantum Jumps, EEQT and the Five Platonic Fractals”, Preprint: http://arXiv.org/abs/quant-ph/0204056. Search in Google Scholar

[4] G. Jastrzebski: “Interacting classical and quantum systems. Chaos from quantum measurements”, Ph.D. thesis (in Polish), University of Wrocław, 1996. Search in Google Scholar

[5] Ö. Stenflo: “Uniqueness of invariant measures for place-dependent random iterations of functions”, IMA Vol. Math. Appl., Vol. 132, (2002), pp. 13–32. (Preprint: http://www.math.su.se/stenflo/IMA.pdf) 10.1007/978-1-4684-9244-6_2Search in Google Scholar

[6] M.F. Barnsley, S.G. Demko, J.H. Elton and J.S. Geronimo: “Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities”, Ann. Inst. H. Poincaré Probab. Statist, Vol. 24, (1988), pp. 367–294. (Erratum: Vol. 25, (1989), pp. 589–590) Search in Google Scholar

[7] A. Jadczyk, G. Kondrat and R. Olkiewicz: “On uniqueness of the jump process in quantum measurement theory”, J. Phys. A, Vol. 30, (1996), pp. 1–18. (Preprinthttp://arXiv.org/abs/quant-ph/9512002) Search in Google Scholar

[8] Ph. Blanchard and A. Jadczyk: “On the Interaction Between Classical and Quantum Systems”, Phys. Lett. A, Vol. 175, (1993), pp. 157–164. (Preprinthttp://arXiv.org/abs/quant-ph/9512002) http://dx.doi.org/10.1016/0375-9601(93)90818-K10.1016/0375-9601(93)90818-KSearch in Google Scholar

[9] A. Jadczyk: “Topics in Quantum Dynamics”, in Proc. First Caribb. School of Math. and Theor. Phys., Saint-Francois-Guadeloupe 1993, Infinite Dimensional Geometry, Noncommutative Geometry, Operator Algebras and Fundamental Interactions, ed. R. Coquereaux et al., World Scientific, Singapore, 1995. (Preprinthttp://arXiv.org/abs/hep-th/9406204) Search in Google Scholar

[10] A. Jadczyk: “IFS Signatures of Quantum States”, IFT Uni Wroclaw, internal report, September 1993. Search in Google Scholar

[11] Ph. Blanchard, A. Jadczyk and R. Olkiewicz: “Completely Mixing Quantum Open Systems and Quantum Fractals”, Physica D: Nonlinear Phenomena, Vol.148, (2001), pp.227–241. (Preprinthttp://arXiv.org/abs/quant-ph/9909085) http://dx.doi.org/10.1016/S0167-2789(00)00175-510.1016/S0167-2789(00)00175-5Search in Google Scholar

[12] A. Lozinski, K. Zyczkowski and W. Slomczynski: “Quantum Iterated Function Systems”, (Phys. Rev., Vol. E68, (2003), article 046110. (Preprinthttp://arXiv.org/abs/quant-ph/0210029) 10.1103/PhysRevE.68.046110Search in Google Scholar PubMed

Published Online: 2004-9-1
Published in Print: 2004-9-1

© 2004 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Scroll Up Arrow