Communications in Mathematical Physics

, Volume 155, Issue 1, pp 71–92 | Cite as

The analogues of entropy and of Fisher's information measure in free probability theory, I

  • Dan Voiculescu
Article

Abstract

Analogues of the entropy and Fisher information measure for random variables in the context of free probability theory are introduced. Monotonicity properties and an analogue of the Cramer-Rao inequality are proved.

Keywords

Entropy Neural Network Statistical Physic Complex System Nonlinear Dynamics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akhiezer, N.I.: The classical moment problem (in Russian) Moscow, 1961Google Scholar
  2. 2.
    Balian, R.: Random matrices and information theory. Nuovo Cimento,LVIIB, No. 1, 183–193 (1968)Google Scholar
  3. 3.
    Barron, A.R.: Entropy and the central limit theorem. Ann. Prob.14(1), 336–342 (1986)Google Scholar
  4. 4.
    Bercovici, H., Voiculescu, D.: Levy-Hinčin type theorems for multiplicative and additive free convolution. Pacific J. Math.153, No. 2, 217–248 (1992)Google Scholar
  5. 5.
    Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Preprint, Berkeley 1992Google Scholar
  6. 6.
    Carlen, E.A., Soffer, A.: Entropy production by block variable summation and central limit theorems. Commun. Math. Phys.140, 339–371 (1991)Google Scholar
  7. 7.
    Duren, P.L.: Univalent functions. Berlin, Heidelberg, New York: Springer Verlag, 1983Google Scholar
  8. 8.
    Dykema, K.J.: On certain free product factors via an extended matrix model. J. Funct. Anal. (to appear)Google Scholar
  9. 9.
    Garnett, J.B.: Bounded analytic functions. New York: Academic Press, 1981Google Scholar
  10. 10.
    Gelfand, I.M., Shilov, G.E.: Generalized functions. I (in Russian). Second edition. Moscow: Fizmatgiz, 1959Google Scholar
  11. 11.
    Kullback, S.: Information theory and statistics. New York: Dover Publications Inc., 1968Google Scholar
  12. 12.
    Landkof, N.S.: Foundations of modern potential theory. Berlin, Heidelberg, New York: Springer 1972Google Scholar
  13. 13.
    Lieb, E.: Convex Trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math.11, 267–288 (1973)Google Scholar
  14. 14.
    Maassen, H.: Addition of freely independent Random variables. J. Funct. Anal.106, 409–438 (1992)Google Scholar
  15. 15.
    Mehta, M.L.: Random matrices and the statistical theory of energy levels. Academic PressGoogle Scholar
  16. 16.
    Mushkhelishvili, N.I.: Singular integral equations. Groningen: Noordhoff 1953Google Scholar
  17. 17.
    Nica, A.: Asymptotically free families of Random unitaries in symmetric groups. Pacific J. Math. (to appear)Google Scholar
  18. 18.
    Popa, S.: Orthogonal pairs of *-subalgebras in finite von Neumann algebras. J. Operator Theory9, 253–268 (1983)Google Scholar
  19. 19.
    Radulescu, F.: The fundamental group of ℒ(F ) is ℝ+\{0}. J. Am. Math. Soc. (to appear)Google Scholar
  20. 20.
    Radulescu, F.: Stable equivalence of the weak closures of free groups convolution algebras. PreprintGoogle Scholar
  21. 21.
    Ruelle, D.: Statistical mechanics. New York: Benjamin 1969Google Scholar
  22. 22.
    Simon, B.: Trace ideals and their applications. Cambridge: Cambridge Univ. Press. 1979Google Scholar
  23. 23.
    Stam, A.J.: Some Inequalities satisfied by the quantities of information of Fisher and Shannon. Information and Control2, 101–112 (1959)Google Scholar
  24. 24.
    Tsuji, M.: Potential Theory in modern function theory. Tokyo: Maruzen 1959Google Scholar
  25. 25.
    Voiculescu, D.: Symmetries of some reduced free productC *-algebras. In: Operator algebras and their connections with topology and ergodic theory. Lecture Notes in Math., vol. 1132. Berlin, Heidelberg, New York: Springer, pp. 556–588 (1985)Google Scholar
  26. 26.
    Voiculescu, D.: Addition of certain non-commuting random variables. J. Funct. Anal.66, No. 3, 323–346 (1986)Google Scholar
  27. 27.
    Voiculescu, D.: Multiplication of certain non-commuting random variables. J. Operator Theory18, 223–235 (1987)Google Scholar
  28. 28.
    Voiculescu, D.: Operations on certain non-commutative operator-valued random variables. INCREST Preprint No. 42/1986, BucharestGoogle Scholar
  29. 29.
    Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math.104, 201–220 (1991)Google Scholar
  30. 30.
    Voiculescu, D.: Circular and semi-circular systems and free product factors. In: Operator algebras, unitary representations, enveloping algebras and invariant theory. Progress in Math., vol. 92, Birkhäuser, Boston, pp. 45–60 (1990)Google Scholar
  31. 31.
    Voiculescu, D.: Free Non-commutative random variables, random matrices and theII 1-factors of free groups. Quantum probability and related topics. VI. Accardi, L. (ed.) Singapore: World Scientific, pp. 473–487 (1991)Google Scholar
  32. 32.
    Wigner, E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math.62, 548–564 (1955)Google Scholar
  33. 33.
    Wigner, E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math.67, 325–327 (1958)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Dan Voiculescu
    • 1
  1. 1.I.H.E.S.Bures-sur-YvetteFrance

Personalised recommendations