Rényi Relative Entropies and Noncommutative $$L_p$$-Spaces II

Abstract

We study an extension of the sandwiched Rényi relative entropies for normal positive functionals on a von Neumann algebra, for parameter values $$\alpha \in [1/2,1)$$ . This work is intended as a continuation of Jenčová (Ann Henri Poincaré 19:2513–2542, 2018), where the values $$\alpha >1$$ were studied. We use the Araki–Masuda divergences of Berta et al. (Ann Henri Poincaré 9:1843–1867, 2018) and treat them in the framework of Kosaki’s noncommutative $$L_p$$ -spaces. Using the variational formula, recently obtained by F. Hiai, for $$\alpha \in [1/2,1)$$ , we prove the data processing inequality with respect to positive trace preserving maps and show that for $$\alpha \in (1/2,1)$$ , equality characterizes sufficiency (reversibility) for any 2-positive trace preserving map.