Skip to main content
SpringerLink
Log in

Quantum Mutual Entropy Defined by Liftings

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

A lifting is a map from the state of a system to that of a compound system, which was introduced in Accardi and Ohya (Appl. Math. Optim. 39:33–59, 1999). The lifting can be applied to various physical processes.

In this paper, we defined a quantum mutual entropy by the lifting. The usual quantum mutual entropy satisfies the Shannon inequality (Ohya in IEEE Trans. Inf. Theory 29(5):770–774, 1983), but the mutual entropy defined through the lifting does not satisfy this inequality unless some conditions hold.

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

Access this article

Log in via an institution

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 expectations, and liftings. Appl. Math. Optim. 39, 33–59 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Accardi, L., Matsuoka, T., Ohya, M.: Entangled Markov chains are indeed entangled. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(3), 379–390 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Gelfand, I.M., Yaglom, A.M.: Calculation of the amount of information about a random function contained in another such function. Am. Math. Soc. Transl. 12, 199–246 (1959)

    MathSciNet  Google Scholar 

  5. Holevo, A.S.: Some estimates for the amount of information transmittable by a quantum communication channel. Probl. Pereda. Inf. 9, 3–11 (1973) (in Russian)

    MATH  MathSciNet  Google Scholar 

  6. Ingarden, R.S.: Quantum information theory. Rep. Math. Phys. 43–73 (1976)

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

    MATH  Google Scholar 

  8. Kolmogorov, A.N.: Theory of transmission of information. Am. Math. Soc. Transl., Ser. 2 33, 291–321 (1963)

    Google Scholar 

  9. Levitin, L.B.: On the quantum measure of information. In: Proceedings of the Fourth Conference on Information Theory, Tashkent, pp. 111–116 (1969). English translation in Annales de la Foundation Louis de Broglie 21(3) (1996)

  10. Ohya, M.: On compound state and mutual information in quantum information theory. IEEE Trans. Inf. Theory 29(5), 770–774 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ohya, M.: Entropy transmission in C*-dynamical systems. J. Math. Anal. Appl. 100(1), 222–235 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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 

  13. Ohya, M.: New quantum algorithm for studying NP-complete problems. Rep. Math. Phys. 52(1), 25–33 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Ohya, M., Petz, D.: Quantum Entropy and Its Use. TMP-Series. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  15. 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 

Download references

Author information

Authors and Affiliations

  1. Department of Information Sciences, Tokyo University of Science, 2641 Yamasaki, Noda City, Chiba, Japan

    Satoshi Iriyama & Masanori Ohya

Authors
  1. Satoshi Iriyama
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masanori Ohya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Satoshi Iriyama.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Iriyama, S., Ohya, M. Quantum Mutual Entropy Defined by Liftings. Found Phys 41, 406–413 (2011). https://doi.org/10.1007/s10701-010-9432-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-010-9432-4

Keywords

Search

Navigation