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Entropy and the Uncertainty Principle

Abstract

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.

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Correspondence to Elliott H. Lieb.

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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.

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Frank, R.L., Lieb, E.H. Entropy and the Uncertainty Principle. Ann. Henri Poincaré 13, 1711–1717 (2012). https://doi.org/10.1007/s00023-012-0175-y

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  • DOI: https://doi.org/10.1007/s00023-012-0175-y

Keywords

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