On the quantum f-relative entropy and generalized data processing inequalities

TL;DR

This paper explores the properties of quantum f-relative entropy, establishing equality conditions for key inequalities and extending these concepts to quantum information measures like Holevo information and entanglement-assisted capacity.

Contribution

It introduces generalized equality conditions for quantum f-relative entropy and extends data processing inequalities to f-analogues of quantum information quantities.

Findings

Equality conditions for monotonicity and joint convexity of quantum f-relative entropy.

Extension of data processing inequality to f-generalized quantum information measures.

New conditions for equality in f-coherent information.

Abstract

We study the fundamental properties of the quantum f-relative entropy, where f(.) is an operator convex function. We give the equality conditions under monotonicity and joint convexity, and these conditions are more general than, since they hold for a class of operator convex functions, and different for f(t) = -ln(t) from, the previously known conditions. The quantum f-entropy is defined in terms of the quantum f-relative entropy and we study its properties giving the equality conditions in some cases. We then show that the f-generalizations of the Holevo information, the entanglement-assisted capacity, and the coherent information also satisfy the data processing inequality, and give the equality conditions for the f-coherent information.