### 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 language | English |
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Pages (from-to) | 205-216 |

Number of pages | 12 |

Journal | Letters in Mathematical Physics |

Volume | 27 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 1993 |

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### 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.

Research output: Contribution to journal › Article

*Letters in Mathematical Physics*, vol. 27, no. 3, pp. 205-216. https://doi.org/10.1007/BF00739578

}

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 -