Metric adjusted skew information

TL;DR

This paper generalizes the concept of skew information in quantum mechanics, introducing metric adjusted skew information that connects quantum statistical geometry with measures of information, and characterizes its mathematical properties.

Contribution

It extends skew information to a broader class called metric adjusted skew information, linking quantum geometry with information measures and identifying its structure.

Findings

Introduces metric adjusted skew information as a non-negative, convex function.

Establishes a connection between quantum statistical geometry and skew information measures.

Characterizes the set of all metric adjusted skew informations as a convex cone generated by lambda-skew informations.

Abstract

We extend the concept of Wigner-Yanase-Dyson skew information to something we call ``metric adjusted skew information'' (of a state with respect to a conserved observable). This ``skew information'' is intended to be a non-negative quantity bounded by the variance (of an observable in a state) that vanishes for observables commuting with the state. We show that the skew information is a convex function on the manifold of states. It also satisfies other requirements, proposed by Wigner and Yanase, for an effective measure-of-information content of a state relative to a conserved observable. We establish a connection between the geometrical formulation of quantum statistics as proposed by Chentsov and Morozova and measures of quantum information as introduced by Wigner and Yanase and extended in this article. We show that the set of normalized Morozova-Chentsov functions describing the possible quantum statistics is a Bauer simplex and determine its extreme points. We determine a particularly simple skew information, the ``lambda-skew information,'' parametrized by a lambda in (0,1], and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations. Key words: Skew information, convexity, monotone metric, Morozova-Chentsov function, lambda-skew information.