Proves integer Rényi QNEC by establishing log-convexity of Kosaki L^n norms under null-translation semigroups for σ-finite von Neumann algebras with half-sided modular inclusions, assuming only finite sandwiched Rényi divergence to the vacuum.
Jenčová,Rényi relative entropies and noncommutativeLp-spaces II,Annales Henri Poincare22(2021) 3235 [1707.00047]
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abstract
We study an extension of the sandwiched R\'enyi 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 [A. Jen\v{c}ov\'a, Ann. Henri Poincar\'e 19, 2513-2542, 2018], where the values $\alpha>1$ were studied. We use the Araki-Masuda divergences of [Berta et al., Ann. Henri Poincar\'e 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.
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