Different quantum f-divergences and the reversibility of quantum operations

TL;DR

This paper systematically compares various quantum $f$-divergences, focusing on their monotonicity and ability to detect reversibility of quantum operations, with implications for quantum information theory.

Contribution

It provides a comprehensive overview of quantum $f$-divergences, analyzing their properties, differences, and conditions under which they indicate reversibility of quantum operations.

Findings

Standard and maximal $f$-divergences differ for non-commuting operators unless $f$ is polynomial.

Monotonicity of $ ext{α-}z$-Rényi divergences depends on parameter domains.

Preservation of certain divergences implies the reversibility of quantum operations.

Abstract

The concept of classical $f$-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the R\'enyi divergences. Various quantum versions of this concept, and more narrowly, the concept of R\'enyi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard $f$-divergences), Matsumoto's maximal $f$-divergences, measured $f$-divergences, and sandwiched and $\alpha$-$z$-R\'enyi divergences. In this paper we give a systematic overview of the various concepts of quantum $f$-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum $f$-divergence by a quantum operation. In particular, we compare the standard and the maximal $f$-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum $f$-divergences are strictly different for non-commuting operators unless $f$ is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard $f$-divergences. We also study the monotonicity of the $\alpha$-$z$-R\'enyi divergences under the special class of bistochastic maps that leave one of the arguments of the R\'enyi divergence invariant, and determine domains of the parameters $\alpha,z$ where monotonicity holds, and where the preservation of the $\alpha$-$z$-R\'enyi divergence implies the reversibility of the quantum operation.