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

D. Petz, Gabor Toth

Research output: Contribution to journalArticle

34 Citations (Scopus)

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
Pages (from-to)205-216
Number of pages12
JournalLetters in Mathematical Physics
Volume27
Issue number3
DOIs
Publication statusPublished - Mar 1993

Fingerprint

Quantum Statistics
quantum statistics
Scalar, inner or dot product
Quantum Systems
monotone functions
Information Inequality
curvature
Totally Geodesic Submanifold
entropy
scalars
Logarithmic Derivative
Riemannian geometry
Relative Entropy
Monotone Function
Sectional Curvature
Scalar Curvature
Density Matrix
products
State Space
Entropy

Keywords

  • Mathematics Subject Classification (1991): 82B10

ASJC Scopus subject areas

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

Cite this

The Bogoliubov inner product in quantum statistics - Dedicated to J. Merza on his 60th birthday. / Petz, D.; Toth, Gabor.

In: Letters in Mathematical Physics, Vol. 27, No. 3, 03.1993, p. 205-216.

Research output: Contribution to journalArticle

@article{f2d750d1e9c64c0bb6fc8a0ebda9a157,
title = "The Bogoliubov inner product in quantum statistics - Dedicated to J. Merza on his 60th birthday",
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 Cram{\'e}r-Rao inequality of classical statistics. These are based on a very general new form of the logarithmic derivative.",
keywords = "Mathematics Subject Classification (1991): 82B10",
author = "D. Petz and Gabor Toth",
year = "1993",
month = "3",
doi = "10.1007/BF00739578",
language = "English",
volume = "27",
pages = "205--216",
journal = "Letters in Mathematical Physics",
issn = "0377-9017",
publisher = "Springer Netherlands",
number = "3",

}

TY - JOUR

T1 - The Bogoliubov inner product in quantum statistics - Dedicated to J. Merza on his 60th birthday

AU - Petz, D.

AU - Toth, Gabor

PY - 1993/3

Y1 - 1993/3

N2 - 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.

AB - 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.

KW - Mathematics Subject Classification (1991): 82B10

UR - http://www.scopus.com/inward/record.url?scp=0010769325&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010769325&partnerID=8YFLogxK

U2 - 10.1007/BF00739578

DO - 10.1007/BF00739578

M3 - Article

AN - SCOPUS:0010769325

VL - 27

SP - 205

EP - 216

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 3

ER -