Understand
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.