A Fundamental Property of Quantum-Mechanical Entropy

  • Elliott H. Lieb
  • Mary Beth Ruskai


There are some properties of entropy, such as concavity and subadditivity, that are known to hold (in classical and in quantum mechanics) irrespective of any assumptions on the detailed dynamics of a system. These properties are consequences of the definition of entropy as S(p) =—Trp lnp (quantum), (1a) S(p) =- f p lnp (classical continuous), (1b) S(p)= p i Inpi (classical discrete), (1c) where Tr means trace, p is a density matrix in (1a), and p is a distribution function (usually on R 6n) in (1b). In (1c) the p i are discrete energy level probabilities.


Density Matrix Pure State Generalize Entropy Physical Review Letter Arbitrary Region 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  • Mary Beth Ruskai
    • 2
  1. 1.Institut des Hautes Etudes ScientifiquesFrance
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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