Letters in Mathematical Physics

, Volume 27, Issue 3, pp 205–216

The Bogoliubov inner product in quantum statistics

Dedicated to J. Merza on his 60th birthday
  • Dénes Petz
  • Gabor Toth
Article

Abstract

A natural Riemannian geometry is defined on the state space of a finite quantum system by means of the Bogoliubov scalar product which is infinitesimally induced by the (nonsymmetric) relative entropy functional. The basic geometrical quantities, including sectional curvatures, are computed for a two-level quantum system. It is found that the real density matrices form a totally geodesic submanifold and the von Neumann entropy is a monotone function of the scalar curvature. Furthermore, we establish information inequalities extending the Cramér-Rao inequality of classical statistics. These are based on a very general new form of the logarithmic derivative.

Mathematics Subject Classification (1991)

82B10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amari, S.,Differential-Geometrical Methods in Statistics, Lecture Notes in Stat. 28, Springer, Berlin, Heidelberg, New York, 1985.Google Scholar
  2. 2.
    Bratteli, O. and Robinson, D. W.,Operator Algebras and Quantum Statistical Mechanics II, Springer, New York, Berlin, Heidelberg, 1981.Google Scholar
  3. 3.
    Cencov, N. N.,Statistical Decision Rules and Optimal Inferences, Amer. Math. Soc., Providence, 1982.Google Scholar
  4. 4.
    Fannes, M. and Verbeure, A., Correlation inequalities and equilibrium states,Comm. Math. Phys. 55, 125–131 (1977).Google Scholar
  5. 5.
    Garrison, J. C. and Wong, J., Bogoliubov inequalities for infinite systems,Comm. Math. Phys. 26, 1–5 (1970).Google Scholar
  6. 6.
    Helstrom, C. W., Minimum mean-square error of estimates in quantum statistics,Phys. Lett. 25, 101–102 (1967).Google Scholar
  7. 7.
    Helstrom, C. W.,Quantum Detection and Estimation Theory, Academic Press, New York, 1976.Google Scholar
  8. 8.
    Hicks, N. J.,Notes on Differential Geometry, Van Nostrand, Princeton, 1965.Google Scholar
  9. 9.
    Holevo, A. S., An analogue of the theory of statistical decisions in non-commutative probability theory,Trans. Moscow Math. Soc. 26, 133–149 (1972).Google Scholar
  10. 10.
    Holevo, A. S.,Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam, 1982.Google Scholar
  11. 11.
    Ingarden, R. S., Janyszek, H., Kossakowski, A., and Kawaguchi, T., Information geometry of quantum statistical systems,Tensor 37, 105–111 (1982).Google Scholar
  12. 12.
    Marshall, A. W. and Olkin, I., Inequalities for the trace function,Aequationes Math. 29, 36–39 (1985).Google Scholar
  13. 13.
    Mori, H., Transport, collective motion and Brownian motion,Progr. Theoret. Phys. 33, 423–455 (1965).Google Scholar
  14. 14.
    Nagaoka, H., A new approach to Cramér-Rao bounds for quantum state estimation, IEICE Technical Report, 89, No. 228, IT89-42, pp. 9-14.Google Scholar
  15. 15.
    Naudts, J., Verbeure, A., and Weder, R., Linear response theory and the KMS condition,Comm. Math. Phys. 44, 87–99 (1975).Google Scholar
  16. 16.
    Petz, D., Entropy in quantum probability I,Quantum Probability and Related Topics VII, World Scientific, Singapore, pp. 275-297.Google Scholar
  17. 17.
    Petz, D. and Toth, G.: work in progress.Google Scholar
  18. 18.
    Umegaki, H., Conditional expectations in an operator algebra IV (entropy and information),Kodai Math. Sem. Rep. 14, 59–85 (1962).Google Scholar
  19. 19.
    Wehrl, A., General properties of entropy,Rev. Modern Phys. 50, 221–260 (1978).Google Scholar
  20. 20.
    Yuen, H. P. and Lax, M., Multiple-parameter quantum estimation and measurement of nonselfadjoint observables,Trans. IEEE IT-19, 740–750 (1973).Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Dénes Petz
    • 1
  • Gabor Toth
    • 2
  1. 1.Department of MathematicsFaculty of Chemical Engineering, Technical University BudapestBudapest XIHungary
  2. 2.Department of Mathematical SciencesRutgers UniversityCamdenUSA

Personalised recommendations