Maps on positive operators preserving Rényi type relative entropies and maximal $f$-divergences

Abstract: In this paper we deal with two quantum relative entropy preserver problems on the cones of positive (either positive definite or positive semidefinite) operators. The first one is related to a quantum R\'enyi relative entropy like quantity which plays an important role in classical-quantum channel decoding. The second one is connected to the so-called maximal $f$-divergences introduced by D. Petz and M. B. Ruskai who considered this quantity as a generalization of the usual Belavkin-Staszewski relative entropy. We emphasize in advance that all the results are obtained for finite dimensional Hilbert spaces.