Communications in Mathematical Physics

, Volume 260, Issue 3, pp 659–671 | Cite as

A Quantum Version of Sanov's Theorem

  • Igor BjelakovićEmail author
  • Jean-Dominique Deuschel
  • Tyll Krüger
  • Ruedi Seiler
  • Rainer Siegmund-Schultze
  • Arleta Szkoła


We present a quantum version of Sanov's theorem focussing on a hypothesis testing aspect of the theorem: There exists a sequence of typical subspaces for a given set Ψ of stationary quantum product states asymptotically separating them from another fixed stationary product state. Analogously to the classical case, the separating rate on a logarithmic scale is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set Ψ. While in the classical case the separating subsets can be chosen universally, in the sense that they depend only on the chosen set of i.i.d. processes, in the quantum case the choice of the separating subspaces depends additionally on the reference state.


Entropy Stationary Quantum Reference State Quantum Computing Logarithmic Scale 
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. Invent. Math. 155(1), 203–222 (2004)CrossRefGoogle Scholar
  2. 2.
    Bjelaković, I., Krüger, T., Siegmund-Schultze, Ra., Szkoła, A.: Chained typical subspaces-A quantum version of Breiman's theorem., 2003
  3. 3.
    Bjelaković, I., Szkoła, A.: The data compression theorem for ergodic quantum information sources. Quant. Inform. Proc. 4(1), 49–63 (2005)CrossRefGoogle Scholar
  4. 4.
    Bjelaković, I., Deuschel, J.-D., Krüger, T., Seiler, R., Siegmund-Schultze, Ra., Szkoła, A.: A Sanov type theorem for ergodic probability measures and its quantum extension. In preparationGoogle Scholar
  5. 5.
    Bjelaković, I., Siegmund-Schultze, Ra.: An Ergodic Theorem for the Quantum Relative Entropy. Commun. Math. Phys. 247, 697–712 (2004)CrossRefGoogle Scholar
  6. 6.
    Bjelaković, I., Siegmund-Schultze, Ra.: A New Proof of the Monotonicity of Quantum Relative Entropy for Finite Dimensional Systems., 2003
  7. 7.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. New York: John Wiley and Sons, 1991Google Scholar
  8. 8.
    Deuschel, J.-D., Stroock, D.W.: Large Deviations. Boston: Acad. Press, 2001Google Scholar
  9. 9.
    Gautschi, W.: Norm estimations for inverses of Vandermonde matrices. Numerische Mathematik, 23, 337–347 (1975)Google Scholar
  10. 10.
    Hiai, F., Petz, D.: The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability. Commun. Math. Phys. 143, 99–114 (1991)CrossRefGoogle Scholar
  11. 11.
    Jozsa, R., Schumacher, B.: A New Proof of the Quantum Noiseless Coding Theorem. J. Mod. Optics 41(12), 2343–2349 (1994)Google Scholar
  12. 12.
    Kaltchenko, A., Yang, E.H.: Universal Compression of Ergodic Quantum Sources. Quant. Inf. and Comput. 3, 359–375 (2003)Google Scholar
  13. 13.
    Ogawa, T., Nagaoka, H.: Strong Converse and Stein's Lemma in Quantum Hypothesis Testing. IEEE Trans. Inf. Th. 46(7), 2428–2433 (2000)Google Scholar
  14. 14.
    Ohya, M., Petz, D.: Quantum entropy and its use. Berlin-Heidelberg-New York: Springer-Verlag, 1993Google Scholar
  15. 15.
    Sanov, I.N.: On the probability of large deviations of random variables. Mat. Sbornik 42, 11–44 (1957)Google Scholar
  16. 16.
    Shields, P.C.: Two divergence-rate counterexamples. J. Theor. Prob. 6, 521–545 (1993)CrossRefGoogle Scholar
  17. 17.
    Takesaki, M.: Theory of operator algebras I. Berlin-Heidelberg-New York: Springer-Verlag, 1979Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Igor Bjelaković
    • 1
    Email author
  • Jean-Dominique Deuschel
    • 1
  • Tyll Krüger
    • 1
    • 2
  • Ruedi Seiler
    • 1
  • Rainer Siegmund-Schultze
    • 1
    • 3
  • Arleta Szkoła
    • 1
    • 4
  1. 1.Institut für Mathematik MA 7-2Technische Universität Berlin, Fakultät II - Mathematik und NaturwissenschaftenBerlinGermany
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  3. 3.Institut für MathematikTechnische Universität IlmenauIlmenauGermany
  4. 4.Max Planck Institute for Mathematics in the Sciences LeipzigGermany

Personalised recommendations