## 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 (US) |
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Pages (from-to) | 205-216 |

Number of pages | 12 |

Journal | Letters in Mathematical Physics |

Volume | 27 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1993 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

## Keywords

- Mathematics Subject Classification (1991): 82B10