Fano type quantum inequalities in terms of $q$-entropies

TL;DR

This paper extends quantum Fano inequalities using $q$-entropies, introducing a $q$-entropy exchange concept and establishing bounds relevant for quantum information theory.

Contribution

It introduces the $q$-entropy exchange and generalizes quantum Fano inequalities for a range of $q$-parameters, expanding theoretical tools in quantum information.

Findings

Established Fano type inequalities with $q$-entropic functionals.

Introduced the $q$-entropy exchange and analyzed its properties.

Derived lower bounds for Tsallis relative $q$-entropy using trace norm.

Abstract

Generalizations of the quantum Fano inequality are considered. The notion of $q$-entropy exchange is introduced. This quantity is concave in each of its two arguments. For $q\geq0$, the inequality of Fano type with $q$-entropic functionals is established. The notion of coherent information and the perfect reversibility of a quantum operation are discussed in the context of $q$-entropies. By the monotonicity property, the lower bound of Pinsker type in terms of the trace norm distance is obtained for the Tsallis relative $q$-entropy of order $q=1/2$. For $0\leq{q}\leq2$, Fano type quantum inequalities with freely variable parameters are obtained.