### Abstract

A Riemannian metric is defined on the state space of a finite quantum system by the canonical correlation (or Kubo-Mori/Bogoliubov scalar product). This metric is infinitesimally induced by the (nonsymmetric) relative entropy functional or the von Neumann entropy of density matrices. Hence its geometry expresses maximal uncertainty. It is proven that the metric is monotone under stochastic mappings, however, an example shows that it is not the only such Riemannian metric. This fact is remarkable because in the probabilistic case, the Markovian monotonicity property characterizes the Fisher information metric. The essential difference appears in the curvatures of a classical state space and a quantum one. A conjecture is made that the scalar curvature is monotone with respect to the "more mixed" (statistical) partial order of density matrices. Furthermore, an information inequality resembling the Cramér-Rao inequality of classical statistics is established. The inequality provides a lower bound for the canonical correlation matrix (of an unbiased observable) and it is saturated when a (partial) observation level and the corresponding family of coarse-grained states are considered.

Original language | English |
---|---|

Pages (from-to) | 780-795 |

Number of pages | 16 |

Journal | Journal of Mathematical Physics |

Volume | 35 |

Issue number | 2 |

Publication status | Published - 1994 |

### Fingerprint

### ASJC Scopus subject areas

- Organic Chemistry

### Cite this

*Journal of Mathematical Physics*,

*35*(2), 780-795.

**Geometry of canonical correlation on the state space of a quantum system.** / Petz, D.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 35, no. 2, pp. 780-795.

}

TY - JOUR

T1 - Geometry of canonical correlation on the state space of a quantum system

AU - Petz, D.

PY - 1994

Y1 - 1994

N2 - A Riemannian metric is defined on the state space of a finite quantum system by the canonical correlation (or Kubo-Mori/Bogoliubov scalar product). This metric is infinitesimally induced by the (nonsymmetric) relative entropy functional or the von Neumann entropy of density matrices. Hence its geometry expresses maximal uncertainty. It is proven that the metric is monotone under stochastic mappings, however, an example shows that it is not the only such Riemannian metric. This fact is remarkable because in the probabilistic case, the Markovian monotonicity property characterizes the Fisher information metric. The essential difference appears in the curvatures of a classical state space and a quantum one. A conjecture is made that the scalar curvature is monotone with respect to the "more mixed" (statistical) partial order of density matrices. Furthermore, an information inequality resembling the Cramér-Rao inequality of classical statistics is established. The inequality provides a lower bound for the canonical correlation matrix (of an unbiased observable) and it is saturated when a (partial) observation level and the corresponding family of coarse-grained states are considered.

AB - A Riemannian metric is defined on the state space of a finite quantum system by the canonical correlation (or Kubo-Mori/Bogoliubov scalar product). This metric is infinitesimally induced by the (nonsymmetric) relative entropy functional or the von Neumann entropy of density matrices. Hence its geometry expresses maximal uncertainty. It is proven that the metric is monotone under stochastic mappings, however, an example shows that it is not the only such Riemannian metric. This fact is remarkable because in the probabilistic case, the Markovian monotonicity property characterizes the Fisher information metric. The essential difference appears in the curvatures of a classical state space and a quantum one. A conjecture is made that the scalar curvature is monotone with respect to the "more mixed" (statistical) partial order of density matrices. Furthermore, an information inequality resembling the Cramér-Rao inequality of classical statistics is established. The inequality provides a lower bound for the canonical correlation matrix (of an unbiased observable) and it is saturated when a (partial) observation level and the corresponding family of coarse-grained states are considered.

UR - http://www.scopus.com/inward/record.url?scp=36449005102&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36449005102&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:36449005102

VL - 35

SP - 780

EP - 795

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 2

ER -