Metric adjusted skew information

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.

Edited the abstract and the introduction

References in corpus (3)

• An elementary formula for entanglement entropies of fermionic systems
• The Wigner-Yanase entropy is not subadditive
• Trace functions as Laplace transforms

Cited by in corpus (10)

• The Wigner-Yanase entropy is not subadditive
• Uncertainty principle with quantum Fisher information
• Inequalities for quantum skew information
• A volume inequality for quantum Fisher information and the uncertainty principle
• Quantum covariance, quantum Fisher information and the uncertainty principle
• A Robertson-type Uncertainty Principle and Quantum Fisher Information
• From quasi-entropy to skew information
• On a correspondence between regular and non-regular operator monotone functions
• Riemannian metrics on positive definite matrices related to means
• An inequality related to uncertainty principle in von Neumann algebras