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Complexity, Fractal Dimension for Quantum States

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Open Systems & Information Dynamics

Abstract

The complexities in information dynamics are reviewed and their examples are given. The fractal dimensions of a quantum state are discussed from a general point of view of complexity. It is shown trough a model that the fractal dimensions of a state provide measures for order structure of chaotic systems.

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Bibliography

  1. L. Accardi, Noncommutative Markov Chains, International School of Mathematical Physics, Camerino, pp. 268-295, 1974.

    Google Scholar 

  2. L. Accardi and M. Ohya, Compound Channels, Transition Expectations and Liftings, to appear in J. Multivariate Analysis.

  3. L. Accardi, M. Ohya and N. Watanabe, Rep. Math. Phys. 38, 1 (1996).

    Google Scholar 

  4. H. Araki, Publ. RIMS, Kyoto Univ. 11, 809 (1976).

    Google Scholar 

  5. F. Benatti, Deterministic Chaos in Infinite Quantum Systems, Trieste Notes in Physics, Springer-Verlag, 1993.

  6. P. Billingsley, Ergodic Theory and Information, Wiley, New York, 1965.

    Google Scholar 

  7. O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, Springer, New York, Berlin, Heidelberg, 1981.

    Google Scholar 

  8. G. J. Chaitin, Algorithmic Information Theory, Cambridge Uni. Press, 1987.

  9. G. Choquet, Lecture Analysis I, II, III, Bengamin, New York, 1969.

    Google Scholar 

  10. A. Connes, H. Narnhofer, and W. Thirring, Commun. Math. Phys. 112, 691 (1987).

    Google Scholar 

  11. A. DeLuca and S. Termini, Inform. Control. 20, 301 (1972).

    Google Scholar 

  12. G. G. Emch, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 29, 241 (1974).

    Google Scholar 

  13. B. R. Ebanks, J. Math. Anal. Appl. 94, 24 (1983).

    Google Scholar 

  14. K. H. Fichtner, W. Freudenberg, and V. Liebscher, Beam Splitting and Time Evolutions of Boson Systems, preprint.

  15. R. S. Ingarden, A. Kossakowski, and M. Ohya, Open Systems and Information Dynamics, to be published in Kluwer.

  16. K. Inoue, T. Matsuoka, and M. Ohya, New Approach to ε-Entropy and Its Comparison with Kolmogolov's ε-entropy, SUT preprint.

  17. A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 119, 861 (1958).

    Google Scholar 

  18. A. N. Kolmogorov, Amer. Math. Soc. Translation, Ser. 2 33, 291 (1963).

    Google Scholar 

  19. S. Kullback and R. Leibler, Ann. Math. Stat. 22, 79 (1951).

    Google Scholar 

  20. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freemann and Company, San Francisco, 1982.

    Google Scholar 

  21. T. Matsuoka and M. Ohya, Rep. Math. Phys. 36, 27 (1995).

    Google Scholar 

  22. T. Matsuoka and M. Ohya, Fractal Dimension of States and Its Application to Shape Analysis Problem, SUT preprint.

  23. N. Muraki, M. Ohya, and D. Petz, Open Sys. Information Dyn. 1, 43 (1992).

    Google Scholar 

  24. N. Muraki and M. Ohya, Lett. Math. Phys. 36, 327 (1996).

    Google Scholar 

  25. J. von Neumann, Die Mathematischen Grundlagen der Quantenmechanik, Springer, Berlin, 1932.

    Google Scholar 

  26. M. Ohya, J. Math. Anal. Appl. 84, 318 (1981).

    Google Scholar 

  27. M. Ohya, L. Nuovo Cimento 38, 402 (1983).

    Google Scholar 

  28. M. Ohya, IEEE Trans. Information Theory 29, 770 (1983).

    Google Scholar 

  29. M. Ohya, J. Math. Anal. Appl. 100, 222 (1984).

    Google Scholar 

  30. M. Ohya, Rep. Math. Phys. 27, 19 (1989).

    Google Scholar 

  31. M. Ohya, Proc. Symp. Appl. Func. Anal. 11, 45 (1989).

    Google Scholar 

  32. M. Ohya, Lecture Notes in Physics 378, Springer, 81 (1991).

    Google Scholar 

  33. M. Ohya, Quantum Probability and Related Topics 6, 359 (1991).

    Google Scholar 

  34. M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer-Verlag, 1993.

  35. M. Ohya and H. Suyari, Rep. Math. Phys. 36, 403 (1995).

    Google Scholar 

  36. M. Ohya, Quantum Communications and Measurement 2, Plenum, 309 (1995).

    Google Scholar 

  37. N. Ohya and N. Watanabe, Note on Irreversible Dynamics and Quantum Information, to appear in the Alberto Frigerio Conference Proceedings.

  38. M. Ohya, Analysis of Geneome Sequences by Complexity, SUT preprint.

  39. N. Ohya and N. Watanabe, On Mathematical Treatment of Fredkin-Toffooli-Milburn Gate, SUT preprint.

  40. C. E. Shannon, Bell System Tech. J. 27, 379 (1948).

    Google Scholar 

  41. A. Uhlmann, Commun. Math. Phys. 54, 21 (1977).

    Google Scholar 

  42. K. Urbanik, Stud. Math. 21, 119 (1961).

    Google Scholar 

  43. L. A. Zadeh, J. Math. Anal. Appl. 23, 421 (1968).

    Google Scholar 

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Ohya, M. Complexity, Fractal Dimension for Quantum States. Open Systems & Information Dynamics 4, 141–157 (1997). https://doi.org/10.1023/A:1009635318043

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