Skip to main content

Entropy and the Uncertainty Principle


We generalize, improve and unify theorems of Rumin, and Maassen–Uffink about classical entropies associated with quantum density matrices. These theorems refer to the classical entropies of the diagonals of a density matrix in two different bases. Thus, they provide a kind of uncertainty principle. Our inequalities are sharp because they are exact in the high-temperature or semi-classical limit.


  1. Beckner W.: Inequalities in Fourier analysis. Ann. Math. (2) 102(1), 159–182 (1975)

    MathSciNet  MATH  Article  Google Scholar 

  2. Carlen, E.A.: Trace inequalities and quantum entropy. An introductory course. In: Sims, R., Ueltschi, D. (eds.) Entropy and the Quantum. Contemporary Mathematics, vol. 529, pp. 73–140. American Mathematical Society, Providence (2010)

  3. Deutsch D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631–633 (1983)

    MathSciNet  ADS  Article  Google Scholar 

  4. Hardy G.H., Littlewood J.E., Pólya G.: Inequalities. Cambridge University Press, Cambridge (1952)

    MATH  Google Scholar 

  5. Hirschman I.I. Jr: A note on entropy. Am. J. Math. 79, 152–156 (1957)

    MathSciNet  MATH  Article  Google Scholar 

  6. Kraus K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987)

    MathSciNet  ADS  Article  Google Scholar 

  7. Lieb E.H., Seiringer R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  8. Maassen H., Uffink J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60(12), 1103–1106 (1988)

    MathSciNet  ADS  Article  Google Scholar 

  9. Rumin M.: Balanced distribution-energy inequalities and related entropy bounds. Duke Math. J. 160(3), 567–597 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  10. Simon, B.: Trace Ideals and Their Applications, 2nd edn. American Mathematical Soceity, Providence (2005)

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Elliott H. Lieb.

Additional information

Copyright © 2012 by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes.

The work was partially supported by NSF Grants PHY-1068285 (R.L.F.) and PHY-0965859 (E.H.L.).

Communicated by Bernard Nienhuis.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Frank, R.L., Lieb, E.H. Entropy and the Uncertainty Principle. Ann. Henri Poincaré 13, 1711–1717 (2012).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Entropy
  • Density Matrix
  • Discrete Fourier Transform
  • Uncertainty Principle
  • Trace Class