From f-Divergence to quantum quasi-entropies and their use

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21 Citations (Scopus)


Csiszár's f-divergence of two probability distributions was extended to the quantum case by the author in 1985. In the quantum setting, positive semidefinite matrices are in the place of probability distributions and the quantum generalization is called quasi-entropy, which is related to some other important concepts as covariance, quadratic costs, Fisher information, Cramér-Rao inequality and uncertainty relation. It is remarkable that in the quantum case theoretically there are several Fisher information and variances. Fisher information are obtained as the Hessian of a quasi-entropy. A conjecture about the scalar curvature of a Fisher information geometry is explained. The described subjects are overviewed in details in the matrix setting. The von Neumann algebra approach is also discussed for uncertainty relation.

Original languageEnglish
Pages (from-to)304-325
Number of pages22
Issue number3
Publication statusPublished - Mar 2010


  • F-divergence
  • Fisher information
  • Monotonicity property
  • Quasi-entropy
  • Relative entropy
  • Uncertainty
  • Von Neumann entropy

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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