Some applications of matrix inequalities in R\'enyi entropy

TL;DR

This paper explores the use of matrix inequalities to derive new bounds on R{\'e}nyi entropy and related quantities, which are crucial in quantum information theory.

Contribution

It introduces novel bounds on R{\'e}nyi entropy and relative entropy using matrix inequalities, advancing the theoretical understanding of these measures.

Findings

Derived new bounds on R{\'e}nyi entropy of type \(\beta\)

Established lower bounds for R{\'e}nyi relative entropy

Enhanced theoretical tools for quantum information measures

Abstract

The R{\'e}nyi entropy is one of the important information measures that generalizes Shannon's entropy. The quantum R{\'e}nyi entropy has a fundamental role in quantum information theory, therefore, bounding this quantity is of vital importance. Another important quantity is R{\'e}nyi relative entropy on which R{\'e}nyi generalization of the conditional entropy, and mutual information are defined based. Thus, finding lower bound for R{\'e}nyi relative entropy is our goal in this paper. We use matrix inequalities to prove new bounds on the entropy of type $\beta$, R{\'e}nyi entropy.