From quasi-entropy to skew information

TL;DR

This paper explores the mathematical relationships between quasi-entropies, quantum covariances, and skew-informations, emphasizing the role of operator monotone functions and extending results to von Neumann algebras.

Contribution

It demonstrates that skew-information is the Hessian of a quasi-entropy and extends inequalities to the setting of von Neumann algebras.

Findings

Skew-information is the Hessian of a quasi-entropy.

Extended inequalities to von Neumann algebra setting.

Clarified the dependence on operator monotone functions.

Abstract

This paper gives an overview about particular quasi-entropies, generalized quantum covariances, quantum Fisher informations, skew-informations and their relations. The point is the dependence on operator monotone functions. It is proven that a skew-information is the Hessian of a quasi-entropy. The skew-information and some inequalities are extended to a von Neumann algebra setting.