Skip to main content
Log in

On Entropy of Quantum Compound Systems

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

We review some notions for general quantum entropies. The entropy of the compound systems is discussed and a numerical computation of the quantum dynamical systems is carried for the noisy optical channel.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Accardi, L., Ohya, M.: Compound channels, transition expectation and liftings. Appl. Math. Optim. 39, 33–59 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Accardi, L., Ohya, M., Watanabe, N.: Dynamical entropy through quantum Markov chain. Open Syst. Inf. Dyn. 4, 71–87 (1997)

    Article  MATH  Google Scholar 

  3. Accardi, L., Ohya, M., Watanabe, N.: Note on quantum dynamical entropies. Rep. Math. Phys. 38, 457–469 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Alicki, R., Fannes, M.: Defining quantum dynamical entropy. Lett. Math. Phys. 32, 75–82 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Araki, H.: Relative entropy for states of von Neumann algebras. Publ. RIMS Kyoto Univ. 11, 809–833 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  6. Benatti, F.: Deterministic Chaos in Infinite Quantum Systems. Springer, Berlin (1993)

    Book  Google Scholar 

  7. Choda, M.: Entropy for extensions of Bernoulli shifts. Ergod. Theory Dyn. Syst. 16(6), 1197–1206 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Connes, A., Narnhoffer, H., Thirring, W.: Dynamical entropy of C*algebras and von Neumann algebras. Commun. Math. Phys. 112, 691–719 (1987)

    Article  MATH  ADS  Google Scholar 

  9. Connes, A., Störmer, E.: Entropy for automorphisms of von Neumann algebras. Acta Math. 134, 289–306 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  10. Emch, G.G.: Positivity of the K-entropy on non-abelian K-flows. Z. Wahrscheinlichkeitstheory verw. Gebiete 29, 241 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fichtner, K.H., Freudenberg, W., Liebscher, V.: Beam splittings and time evolutions of Boson systems, Fakultat fur Mathematik und Informatik, Math/Inf/96/39, Jena, 105 (1996)

  12. Hudetz, T.: Topological entropy for appropriately approximated C*-algebras. J. Math. Phys. 35(8), 4303–4333 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Book  MATH  Google Scholar 

  14. Jamiołkowski, A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3, 275–278 (1972)

    Article  MATH  ADS  Google Scholar 

  15. Khrennikov, A.: Contextual Approach to Quantum Formalism. Series of fundamental theories of physics. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  16. Khrennikov, A.: A classical field theory comeback? The classical field viewpoint on triparticle entanglement. Phys. Scripta, T143, Article Number: 014013 (2011). doi:10.1088/0031-8949/2011/T143/014013

  17. Kolmogorov, A.N.: Theory of transmission of information. Am. Math. Soc. Transl. Ser. 2 33, 291 (1963)

    MATH  Google Scholar 

  18. von Neumann, J.: Die Mathematischen Grundlagen der Quantenmechanik. Springer, Berlin (1932)

    Google Scholar 

  19. Kossakowski, A., Ohya, M., Watanabe, N.: Quantum dynamical entropy for completely positive map. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2(2), 267–282 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ohya, M.: Quantum ergodic channels in operator algebras. J. Math. Anal. Appl. 84, 318–328 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ohya, M.: On compound state and mutual information in quantum information theory. IEEE Trans. Inf. Theory 29, 770–774 (1983)

    Article  MATH  Google Scholar 

  22. Ohya, M.: Note on quantum probability. L. Nuovo Cimento 38, 402–404 (1983)

    Article  MathSciNet  Google Scholar 

  23. Ohya, M.: Some aspects of quantum information theory and their applications to irreversible processes. Rep. Math. Phys. 27, 19–47 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Ohya, M.: State change, complexity and fractal in quantum systems. Quantum Commun. Meas. 2, 309–320 (1995)

    Article  MathSciNet  Google Scholar 

  25. Ohya, M., Petz, D.: Quantum Entropy and Its Use. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  26. Ohya, M., Watanabe, N.: Foundation of Quantum Communication Theory (in Japanese). Makino Publishing Company, Tokyo (1998)

    Google Scholar 

  27. Ohya, M., Watanabe, N.: Construction and analysis of a mathematical model in quantum communication processes. Electron. Commun. Jpn. Part 1 68(2), 29–34 (1985)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  29. Park, Y.M.: Dynamical entropy of generalized quantum Markov chains. Lett. Math. Phys. 32, 63–74 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Schatten, R.: Norm Ideals of Completely Continuous Operators. Springer, Berlin (1970)

    Book  MATH  Google Scholar 

  31. Tuyls, P.: Comparing quantum dynamical entropies. Banach Centre Publ. 43, 411–420 (1998)

    MathSciNet  Google Scholar 

  32. Umegaki, H.: Conditional expectations in an operator algebra IV (entropy and information). Kodai Math. Sem. Rep. 14, 59–85 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  33. Uhlmann, A.: Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in interpolation theory. Commun. Math. Phys. 54, 21–32 (1977)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. Voiculescu, D.: Dynamical approximation entropies and topological entropy in operator algebras. Commun. Math. Phys. 170, 249–281 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  35. Watanabe, N.: Note on entropies of quantum dynamical systems. Found. Phys. 41, 549–563 (2011). doi:10.1007/s10701-010-9455-x

    Article  MATH  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Noboru Watanabe.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Watanabe, N. On Entropy of Quantum Compound Systems. Found Phys 45, 1311–1329 (2015). https://doi.org/10.1007/s10701-015-9925-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-015-9925-2

Keywords

Navigation