Upper continuity bounds on the relative $q$-entropy for $q>1$

TL;DR

This paper establishes upper continuity bounds for the quantum relative q-entropy when q>1, demonstrating how it can be controlled by norm distances between quantum states in finite dimensions.

Contribution

It provides the first finite-dimensional upper bounds on the quantum relative q-entropy for q>1, extending the Fannes continuity property to this generalized entropy.

Findings

Derived upper bounds relate relative q-entropy to state distances

Bounds are valid in finite-dimensional quantum systems

Results extend continuity properties to generalized entropies

Abstract

Generalized entropies and relative entropies are the subject of active research. Similar to the standard relative entropy, the relative $q$-entropy is generally unbounded for $q>1$. Upper bounds on the quantum relative $q$-entropy in terms of norm distances between its arguments are obtained in finite-dimensional context. These bounds characterize a continuity property in the sense of Fannes.