Letters in Mathematical Physics

, Volume 27, Issue 3, pp 205–216

The Bogoliubov inner product in quantum statistics

Dedicated to J. Merza on his 60th birthday


  • Dénes Petz
    • Department of MathematicsFaculty of Chemical Engineering, Technical University Budapest
  • Gabor Toth
    • Department of Mathematical SciencesRutgers University

DOI: 10.1007/BF00739578

Cite this article as:
Petz, D. & Toth, G. Lett Math Phys (1993) 27: 205. doi:10.1007/BF00739578


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.

Mathematics Subject Classification (1991)


Copyright information

© Kluwer Academic Publishers 1993