Foundations of Physics

, Volume 44, Issue 10, pp 1107–1123 | Cite as

Characterizing Entropy in Statistical Physics and in Quantum Information Theory

  • Bernhard Baumgartner


A new axiomatic characterization with a minimum of conditions for entropy as a function on the set of states in quantum mechanics is presented. Traditionally unspoken assumptions are unveiled and replaced by proven consequences of the axioms. First the Boltzmann–Planck formula is derived. Building on this formula, using the Law of Large Numbers—a basic theorem of probability theory—the von Neumann formula is deduced. Axioms used in older theories on the foundations are now derived facts.


Entropy Axiomatic Information Large numbers 

Mathematics Subject Classification




The author thanks the referees for important hints, and he gives many thanks to Elliott Lieb and Jakob Yngvason for discussions.


  1. 1.
    Åberg, J.: Truly work-like work extraction via a single-shot analysis. Nat. Commun. 4, 1925 (2013)CrossRefGoogle Scholar
  2. 2.
    Aczél, J., Forte, B., Ng, C.T.: Why the Shannon and Hartley entropies are ‘Natural’. Adv. Appl. Probab. 6, 131–146 (1974)CrossRefzbMATHGoogle Scholar
  3. 3.
    Anders, J., Shabbir, S., Hilt, S., Lutz, E.: Landauer’s principle in the quantum domain. Electron. Proc. Theor. Comput. Sci. 26, 13–18 (2010)CrossRefGoogle Scholar
  4. 4.
    Bhatia, R.: Matrix Analysis. Springer, New York (1997)Google Scholar
  5. 5.
    Boltzmann, L.: Analytischer Beweis des zweiten Hauptsatzes der mechanischen Wärmetheorie aus den Sätzen über das Gleichgewicht der lebendigen Kraft. Sitzb. der Wiener Akad. LXIII, 712–732 (1871)Google Scholar
  6. 6.
    Boltzmann, L.: Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzb. der Wiener Akad. LXVI, 275–370 (1872)Google Scholar
  7. 7.
    Boltzmann, L.: Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht. Sitzb. der Wiener Akad. LXXVI, 373–435 (1877)Google Scholar
  8. 8.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. Springer, New York (1981)CrossRefzbMATHGoogle Scholar
  9. 9.
    Brönnimann, D.: Die Entwicklung des Wahrscheinlichkeitsbegriffs von 1654 bis 1718. Broennimann-WahrschkeitGoogle Scholar
  10. 10.
    Cohen, E.G.D., Thirring, W.: The Boltzmann Equation. Springer, Wien (1973)Google Scholar
  11. 11.
    Cohen-Tannoudji, C., Guéry-Odelin, D.: Advances in Atomic Physics. World Scientific, Singapore (2011)Google Scholar
  12. 12.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley Interscience, Hoboken (2006)zbMATHGoogle Scholar
  13. 13.
    Dahlsten, O.C.O., Renner, R., Rieper, E., Vedral, V.: Inadequacy of von Neumann entropy for characterizing extractable work. New J. Phys. 13(5), 053015 (2011), arXiv:0908.0424
  14. 14.
    Dupuis, F., et al.: Generalized entropies. In: Proceedings of the XVIIth International Conference on Mathematics and Physics, Aalborg, Denmark (2012). arXiv:1211.3141
  15. 15.
    Egloff, D., et al.: Laws of thermodynamics beyond the von Neumann regime. arXiv:1207.0434
  16. 16.
    Einstein, A.: Beiträge zur Quantentheorie. Verh. Deutsch. Phys. Ges. 12, 820–828 (1914)Google Scholar
  17. 17.
    Gallavotti, G., Reiter, W.L., Yngvason, J. (eds.).: Boltzmann’s Legacy. European Mathematical Society Publ. House (2008)Google Scholar
  18. 18.
    Gibbs, J.W.: Elementary Principles in Statistical Mechanics. Yale University Press (1902), republicated by Dover Publ., Inc. N.Y. (1960)Google Scholar
  19. 19.
    Ingarden, R.S., Kossakowski, A., Ohya, M.: Information Dynamics and Open Systems. Kluwer Academic Publishers (1997)Google Scholar
  20. 20.
    Klein, M.J.: The Development of Boltzmann’s Statistical Ideas. In: Cohen, E.G.D., Thirring, W. (eds.) The Boltzmann Equation. Springer, Wien (1973)Google Scholar
  21. 21.
    Lieb, E.H., Yngvason, J.: A guide to entropy and the second law of thermodynamics. Not. Am. Math. Soc. 45, 571–581 (1999), Erratum 314, 699 (1999)Google Scholar
  22. 22.
    Lieb, E.H., Yngvason, J.: The physics and mathematics of the second law of thermodynamics. Phys. Rep. 310, 1–96 (1999), Erratum 314, 699 (1999)Google Scholar
  23. 23.
    Lieb, E.H., Yngvason, J.: A fresh look at entropy and the second law of thermodynamics. Phys. Today 310, 32–37 (2000)Google Scholar
  24. 24.
    Lieb, E.H., Yngvason, J.: The Mathematical Structure of the Second Law of Thermodynamics. arXiv:math-ph/0204007Google Scholar
  25. 25.
    Nagaoka, H., Hayashi, M.: An information-spectrum approach to classical and quantum hypothesis testing for simple hypotheses. IEEE Trans. Inf. Theory 53(2), 534–549 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge Inversity Press, Cambridge, UK (2000)zbMATHGoogle Scholar
  27. 27.
    Ochs, W.: A new axiomatic characterization of the von Neumann entropy. Rep. Math. Phys. 3, 109–120 (1975)MathSciNetCrossRefADSGoogle Scholar
  28. 28.
    Planck, M.: Über das Gesetz der Energieverteilung im Normalspektrum. Ann. Phys. 4(4), 553–563 (1901)CrossRefzbMATHGoogle Scholar
  29. 29.
    Planck, M.: Vorlesungen über die Theorie der Wärmestrahlung. Barth, Leipzig (1906)zbMATHGoogle Scholar
  30. 30.
    Renner, R.: Security of quantum key distribution. Dissertation. arXiv:0512258v2Google Scholar
  31. 31.
    Ruskai, M.B.: Inequalities for quantum entropy: a review with conditions for equality. J. Math. Phys. 43, 4358–4375 (2002)Google Scholar
  32. 32.
    Schrödinger, E.: Statistical Thermodynamics. Cambridge University Press, London (1946)zbMATHGoogle Scholar
  33. 33.
    Shannon, C.F.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656, July, October (1948)Google Scholar
  34. 34.
    Shannon, C.F., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Chicago (1949)zbMATHGoogle Scholar
  35. 35.
    Szilard, L.: Über die Ausdehnung der phänomenologischen Thermodynamik auf die Schwankungserscheinungen. Zeitschrift für Physik. XXXII(10), 753–788 (1925)Google Scholar
  36. 36.
    Tomamichel, M.: A Framework for Non-Asymptotic Quantum Information Theory. Dissertation, arXiv:1203.2142
  37. 37.
    Tomamichel, M., Colbeck, R., Renner, R.: A fully quantum asymptotic equipartition property. IEEE Trans. Inf. Theory 55(12), 5840–5847 (2009)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Uhlmann, A.: Sätze über Dichtematrizen. Wiss. Z. Karl-Marx-Univ. 20, 633–637 (1971)MathSciNetGoogle Scholar
  39. 39.
    Uhlmann, A.: Wiss. Z. Karl-Marx-Univ. Endlich-dimensionale Dichtematrizen I 21, 421–452 (1972)MathSciNetGoogle Scholar
  40. 40.
    Uhlmann, A.: Wiss. Z. Karl-Marx-Univ. Endlich-dimensionale Dichtematrizen II 22, 139–177 (1973)MathSciNetGoogle Scholar
  41. 41.
    von Neumann, J.: Gött. Nachr. Thermodynamik quantenmechanischer Gesamtheiten 1, 273–291 (1929)Google Scholar
  42. 42.
    von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932)zbMATHGoogle Scholar
  43. 43.
    Wehrl, A.: How chaotic is a state of a quantum system? Rep. Math. Phys. 6, 15–28 (1974)MathSciNetCrossRefADSGoogle Scholar
  44. 44.
    Wehrl, A.: General properties of entropy. Rev. Mod. Phys. 50, 221–260 (1978)MathSciNetCrossRefADSGoogle Scholar
  45. 45.
    Wehrl, A.: Information-Theoretical Aspects of Quantum-Mechanical Entropy. Univ. Vienna preprint UWThPh-1990-20, unpublished (1990)Google Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität WienViennaAustria

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