Inequalities for quantum skew information

TL;DR

This paper explores inequalities related to quantum skew information, revealing a foundational inequality that generates various matrix inequalities and introducing an order structure that leads to new insights and bounds in quantum information theory.

Contribution

It introduces a novel order relation on quantum Fisher information functions, creating a lattice structure that yields new inequalities and characterizes maximal skew information.

Findings

The basic inequality underpins multiple matrix inequalities.

A lattice structure on quantum Fisher information functions is established.

Wigner-Yanase skew information is identified as maximal within its class.

Abstract

We study quantum information inequalities and show that the basic inequality between the quantum variance and the metric adjusted skew information generates all the multi-operator matrix inequalities or Robertson type determinant inequalities studied by a number of authors. We introduce an order relation on the set of functions representing quantum Fisher information that renders the set into a lattice with an involution. This order structure generates new inequalities for the metric adjusted skew informations. In particular, the Wigner-Yanase skew information is the maximal skew information with respect to this order structure in the set of Wigner-Yanase-Dyson skew informations. Key words and phrases: Quantum covariance, metric adjusted skew information, Robertson-type uncertainty principle, operator monotone function, Wigner-Yanase-Dyson skew information.