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Quantum transmission processes and their entropies

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Abstract

In classical information theory, one of the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov-Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes-Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy are done [10, 3, 8, 39, 15, 44, 9, 27, 28, 2, 19, 45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S- mixing entropy by Ohya [26, 27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S- mixing entropy. In this paper, we discuss the efficiency of information transmission to calculate the mean mutual entropy with respect to the modulated initial states and the connected channel for the quantum dynamical systems.

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References

  1. L. Accardi, and M. Ohya, “Compond Channnels, Transition Expectation and Liftings,” Appl. Math, Optim. 39, 33–59 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Accardi, M. Ohya, and N. Watanabe, “Dynamical Entropy Through Quantum Markov Chain,” Open System and Information Dynamics 4, 71–87 (1997).

    Article  MATH  Google Scholar 

  3. R. Alicki and M. Fannes, “Defining Quantum Dynamical Entropy,” 32, 75–82 (1994).

    MATH  MathSciNet  Google Scholar 

  4. H. Araki, “Relative Entropy for States of von Neumann Algebras,” Publ. RIMS Kyoto Univ. 11, 809–833 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  5. H. Araki, “Relative Entropy for States of von Neumann Algebras II,” 13, 173–192 (1977).

    MATH  MathSciNet  Google Scholar 

  6. V. P. Belavkin, and M. Ohya, “Quantum Entropy and Information in Discrete Entangled States,” Infinite Dimensional Analysis, Quantum Probability and Related Topics 4(2), 137–160 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  7. V.P. Belavkin, and M. Ohya, “Entanglement, Quantum Entropy and Mutual Information,” Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 458, 209–231 (2002).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. F. Benatti, Trieste Notes in Phys (Springer-Verlag, 1993).

    Google Scholar 

  9. M. Choda, “Entropy for Extensions of Bernoulli Shifts,” 16, 1197–1206 (1996).

    MATH  MathSciNet  Google Scholar 

  10. A. Connes, H. Narnhofer, and W. Thirring, “Dynamical Entropy of C*-Algebras and von Neumann Algebras,” 112, 691–719 (1987).

    MATH  MathSciNet  Google Scholar 

  11. A. Connes and E. Stormer, “Entropy for Automorphisms of Von Neumann Algebras,” Acta Math. 134, 289–306 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  12. M. J. Donald, “On the Relative Entropy,” 105, 13–34 (1985).

    MathSciNet  Google Scholar 

  13. G. G. Emch, “Positivity of the K-Entropy on Non-Abelian K-Flows,” Z. Wahrscheinlichkeitstheory und Verw. Gebiete 29(241), (1974).

    Google Scholar 

  14. K. H. Fichtner, W. Freudenberg, and V. Liebscher, “Beam Splittings and Time Evolutions of Boson Systems,” Fakultät für Mathematik und Informatik, Math. Inf. 96/39, Jena, 105 (1996).

    Google Scholar 

  15. T. Hudetz, “Topological Entropy for Appropriately Approximated C*-Algebras,” 35, 4303–4333 (1994).

    MATH  MathSciNet  Google Scholar 

  16. R. S. Ingarden, A. Kossakowski, and M. Ohya, Information Dynamics and Open Systems (Kluwer, 1997).

    Book  MATH  Google Scholar 

  17. A. N. Kolmogorov, “Theory of Transmission of Information,” Ser.2 33, 291–321 (1963).

    MATH  Google Scholar 

  18. A. Kossakowski and M. Ohya, “New Scheme of Quantum Teleportation, Infinite Dimensional Analysis,” Quantum Probability and Related Topics 10(3), 411–420 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Kossakowski, M. Ohya and N. Watanabe, “Quantum Dynamical Entropy for Completely Positive Map,” Infinite Dimensional Analysis, Quantum Probability and Related Topics 2(2), 267–282 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  20. N. Muraki, and M. Ohya, “Entropy Functionals of Kolmogorov Sinai Type and Their Limit Theorems,” Letter in Math. Phys. 36, 327–335 (1996).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. N. Muraki, M. Ohya, and D. Petz, “Entropies of General Quantum States,” Open Systems and Information Dynamics 1, 43–56 (1992).

    Article  MATH  Google Scholar 

  22. J. von Neumann, Die Mathematischen Grundlagen der Quantenmechanik (Springer-Berlin, 1932).

    Google Scholar 

  23. M. Ohya, “Quantum Ergodic Channels in Operator Algebras,” J. Math. Anal. Appl. 84, 318–328 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  24. M. Ohya, “On Compound State and Mutual Information in Quantum Information Theory,” IEEE Trans. Information Theory 29, 770–774 (1983).

    Article  MATH  Google Scholar 

  25. M. Ohya, “Note on Quantum Probability,” L. Nuovo Cimento 38, 402–404 (1983).

    Article  MathSciNet  Google Scholar 

  26. M. Ohya, “Entropy Transmission in C*-Dynamical Systems,” J. Math. Anal. Appl. 100(1), 222–235 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Ohya, “State Change and Entropies in Quantum Dynamical Systems,” Springer Lecture Note in Math. 1136, 397–408 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  28. M. Ohya, “Some Aspects of Quantum Information Theory and Their Applications to Irreversible Processes,” Rep. Math. Phys. 27, 19–47 (1989).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. M. Ohya, “Information Dynamics and Its Application to Optical Communication Processes,” Springer Lecture Note in Physics 378, 81–92 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  30. M. Ohya, “Fundamentals of Quantum Mutual Entropy and Capacity,” 6, 69–78 (1999).

    MATH  MathSciNet  Google Scholar 

  31. M. Ohya, and D. Petz, Quantum Entropy and its Use (Springer, Berlin, 1993).

    Book  MATH  Google Scholar 

  32. M. Ohya, D. Petz, and N. Watanabe, “On Capacity of Quantum Channels,” Probability and Mathematical Statistics 17, 179–196 (1997).

    MATH  MathSciNet  Google Scholar 

  33. M. Ohya, D. Petz, and N. Watanabe, “Numerical Computation of Quantum Capacity,” International Journal of Theoretical Physics 37(1), 507–510 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  34. M. Ohya and I. V. Volovich, “On Quantum Entropy and its Bound,” 6, 301–310 (2003).

    MATH  MathSciNet  Google Scholar 

  35. M. Ohya, and I. Volovich, Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems (Springer, 2011).

    Book  MATH  Google Scholar 

  36. M. Ohya, and N. Watanabe, “Construction and Analysis of a Mathematical Model in Quantum Communication Processes,” Electronics and Communications in Japan, Part 1 68(2), 29–34 (1985).

    Article  Google Scholar 

  37. M. Ohya, and N. Watanabe, “Quantum Capacity of Noisy Quantum Channel,” Quantum Communication and Measurement 3, 213–220 (1997).

    Article  Google Scholar 

  38. M. Ohya, and N. Watanabe, Foundation of Quantum Communication Theory (in Japanese) (Makino Pub. Co., 1998).

    Google Scholar 

  39. Y. M. Park, “Dynamical Entropy of Generalized Quantum Markov Chains,” 32, 63–74 (1994).

    MATH  Google Scholar 

  40. R. Schatten, Norm Ideals of Completely Continuous Operators (Springer-Verlag, 1970).

    Book  MATH  Google Scholar 

  41. C. E. Shannon, “A Mathematical Theory of Communication,” 27, 379–423 and 623–656.

  42. A. Uhlmann, “Relative Entropy and the Wigner-Yanase-Dyson-Lieb Concavity in Interpolation Theory,” Commun. Math. Phys. 54, 21–32 (1977).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  43. H. Umegaki, “Conditional Expectations in an Operator Algebra IV (Entropy and Information),” Kodai Math. Sem. Rep. 14, 59–85 (1962).

    Article  MATH  MathSciNet  Google Scholar 

  44. D. Voiculescu, “Dynamical Approximation Entropies and Topological Entropy in Operator Algebras,” 170, 249–281 (1995).

    MATH  MathSciNet  Google Scholar 

  45. N. Watanabe, Some Aspects of Complexities for Quantum Processes, Open Systems and Information Dynamics,” 16, 293–304 (2009).

    Article  MATH  Google Scholar 

  46. N. Watanabe, “Note on Entropies of Quantum Dynamical Systems,” Foundation of Physics 41, 549–563 1007/s10701-010-9455-x (2011).

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Information Sciences, Tokyo University of Science, Noda City, Chiba, 278-8510, Japan

    N. Watanabe

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  1. N. Watanabe
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Correspondence to N. Watanabe.

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Dedicated to Professor Viacheslav P. Belavkin

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Watanabe, N. Quantum transmission processes and their entropies. Russ. J. Math. Phys. 21, 408–420 (2014). https://doi.org/10.1134/S1061920814030133

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  • DOI: https://doi.org/10.1134/S1061920814030133

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