# Noncommutative versions of inequalities in quantum information theory

## TL;DR

This paper generalizes key inequalities in quantum information theory by replacing trace with positive maps and specific functions, leading to broader noncommutative uncertainty relations and properties.

## Contribution

It introduces a framework for noncommutative inequalities using positive maps and functions, extending classical quantum information inequalities.

## Key findings

- Generalized covariance, variance, and correlation defined.
- A noncommutative version of Heisenberg's uncertainty relation established.
- Extended inequalities for generalized skew information and variance.

## Abstract

In this paper, we aim to replace in the definitions of covariance and correlation the usual trace {\rm Tr} by a tracial positive map between unital $C^*$-algebras and to replace the functions $x^{\alpha}$ and $x^{1-\alpha}$ by functions $f$ and $g$ satisfying some mild conditions. These allow us to define the generalized covariance, the generalized variance, the generalized correlation and the generalized Wigner--Yanase--Dyson skew information related to the tracial positive maps and functions $f$ and $g$. We persent a generalization of Heisenberg's uncertainty relation in the noncommutative framework. We extend some inequalities and properties for the generalized correlation and the generalized Wigner--Yanase--Dyson skew information. Furthermore, we extend some inequalities for the generalized skew information such as uncertainty relation and the relation between the generalized variance and the generalized skew information.