Communications in Mathematical Physics

, Volume 247, Issue 3, pp 697–712 | Cite as

An Ergodic Theorem for the Quantum Relative Entropy

  • Igor Bjelaković
  • Rainer Siegmund-SchultzeEmail author


We prove the ergodic version of the quantum Stein’s lemma which was conjectured by Hiai and Petz. The result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability, and contains as a special case the quantum version of the Shannon-McMillan theorem for ergodic states. A version of the quantum relative Asymptotic Equipartition Property (AEP) is given.


Entropy Stein Statistical Measure Relative Entropy Ergodic Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bjelaković, I., Krüger, T., Siegmund-Schultze, Ra., Szkoła, A.: The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems., math.DS/0207121. Invent. Math. 155, 203–222 (2004)Google Scholar
  2. 2.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. New York: Springer, 1979Google Scholar
  3. 3.
    Hiai, F., Petz, D.: The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability. Commun. Math. Phys. 143, 99–114 (1991)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ogawa, T., Nagaoka, H.: Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing. IEEE Trans. Inform. Theo. 46(7), 2428–2433 (2000)zbMATHGoogle Scholar
  5. 5.
    Ohya, M., Petz, D.: Quantum Entropy and its Use. Berlin: Springer, 1993Google Scholar
  6. 6.
    Ornstein, D., Weiss, B.: The Shannon-McMillan-Breiman Theorem for a Class of Amenable Groups. Israel J. Math. 44(1), 53–60 (1983)zbMATHGoogle Scholar
  7. 7.
    Ruelle, D.: Statistical Mechanics. New York: W.A. Benjamin, 1969Google Scholar
  8. 8.
    Shields, P.C.: Two divergence-rate counterexamples. J. Theor. Prob. 6, 521–545 (1993)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Technische Universität BerlinFakultät II - Mathematik und Naturwissenschaften, Institut für Mathematik MA 7-2BerlinGermany

Personalised recommendations