Quantum statistics and measures of quantum information

Frank Hansen

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

The geometrical formulation of quantum statistics and the theory of measures of quantum information are intimately connected by the introduction of the notion of metric adjusted skew information.We survey the area with a focus on the ‘representing operator monotone functions’. In particular, we exhibit a new order structure that renders the set of metric adjusted skew informations into a lattice with a maximal element. The Wigner-Yanase-Dyson skew informations (with parameter p) are increasing with respect to this order structure for 0 < p ≤ 1/2 and decreasing for 1/2 ≤ p < 1 with maximum in the Wigner-Yanase skew information. Introduction The geometrical formulation of quantum statistics and the theory of measures of quantum information are two distinct theories with separate motivations and histories, and for a long time they did not seem to be related in any way. Early contributions which view quantum information in statistical terms were given by Hasegawa and Petz (Hasegawa and Petz 1996), and by Luo (Luo 2003a, Luo 2003b, Luo 2005). Today the two theories are largely fused with the notion of metric (or metrically) adjusted skew information and a common set of tools and techniques involving certain operator monotone functions and their representations. The geometrical approach to statistics reveals its fundamental nature when we try to generalise classical statistics to the quantum setting. The key to obtaining quantisation of the Fisher information is to consider it as a Riemannian metric with a certain behaviour in the presence of noise.

Original languageEnglish
Title of host publicationAlgebraic and Geometric Methods in Statistics
PublisherCambridge University Press
Pages327-338
Number of pages12
ISBN (Electronic)9780511642401
ISBN (Print)9780521896191
DOIs
Publication statusPublished - 2009 Jan 1

ASJC Scopus subject areas

  • Mathematics(all)

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