The Bogoliubov inner product in quantum statistics - Dedicated to J. Merza on his 60th birthday

Dénes Petz, Gabor Toth

Research output: Contribution to journalArticle

36 Scopus citations

Abstract

A natural Riemannian geometry is defined on the state space of a finite quantum system by means of the Bogoliubov scalar product which is infinitesimally induced by the (nonsymmetric) relative entropy functional. The basic geometrical quantities, including sectional curvatures, are computed for a two-level quantum system. It is found that the real density matrices form a totally geodesic submanifold and the von Neumann entropy is a monotone function of the scalar curvature. Furthermore, we establish information inequalities extending the Cramér-Rao inequality of classical statistics. These are based on a very general new form of the logarithmic derivative.

Original languageEnglish (US)
Pages (from-to)205-216
Number of pages12
JournalLetters in Mathematical Physics
Volume27
Issue number3
DOIs
StatePublished - Mar 1 1993

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

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