2005/04/30 by Paul B. Slater
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Complex Systems and Time Series Analysis #Convex metric space #Discrete mathematics #Fisher information #Fisher information metric #Kullback–Leibler divergence #Mathematics #Metric space #Physics #Probability distribution #Pure mathematics #Quantum #Quantum Mechanics and Applications #Quantum mechanics #Statistical Mechanics and Entropy #Statistical distance #Statistical physics #Statistics #quant-ph
paper · pdf · doi:10.1063/1.2168125
published as J. Math. Phys. 47, 022104 (2006) (17 pages) · 27 pages, 10 figures, slight revisions, to appear in J. Math. Phys
arxiv created 2006/01/02 · openalex publication_date 2006/02/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The two principal/immediate influences—which we seek to interrelate here—upon the undertaking of this study are papers of Życzkowski and Słomczyński [J. Phys. A 34, 6689 (2001)] and of Petz and Sudár [J. Math. Phys. 37, 2262 (1996)]. In the former work, a metric (the Monge one, specifically) over generalized Husimi distributions was employed to define a distance between two arbitrary density matrices. In the Petz-Sudár work (completing a program of Chentsov), the quantum analog of the (classically unique) Fisher information (monotone) metric of a probability simplex was extended to define an uncountable infinitude of Riemannian (also monotone) metrics on the set of positive definite density matrices. We pose here the questions of what is the specific/unique Fisher information metric for the (classically defined) Husimi distributions and how does it relate to the infinitude of (quantum) metrics over the density matrices of Petz and Sudár? We find a highly proximate (small relative entropy) relationship between the probability distribution (the quantum Jeffreys’ prior) that yields quantum universal data compression, and that which (following Clarke and Barron) gives its classical counterpart. We also investigate the Fisher information metrics corresponding to the escort Husimi, positive-P and certain Gaussian probability distributions, as well as, in some sense, the discrete Wigner pseudoprobability. The comparative noninformativity of prior probability distributions—recently studied by Srednicki [Phys. Rev. A 71, 052107 (2005)]—formed by normalizing the volume elements of the various information metrics, is also discussed in our context.