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

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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 languageEnglish
Pages (from-to)780-795
Number of pages16
JournalJournal of Mathematical Physics
Volume35
Issue number2
Publication statusPublished - 1994

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Canonical Correlation
Quantum Systems
State Space
Density Matrix
Riemannian Metric
Metric
Geometry
Monotone
Entropy
geometry
Information Inequality
curvature
entropy
scalars
Partial Observation
Fisher information
Fisher Information
Relative Entropy
Correlation Matrix
Scalar Curvature

ASJC Scopus subject areas

  • Organic Chemistry

Cite this

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

In: Journal of Mathematical Physics, Vol. 35, No. 2, 1994, p. 780-795.

Research output: Contribution to journalArticle

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