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arXiv · 1707.00047

R\'enyi relative entropies and noncommutative $L_p$-spaces II

Anna Jenčová
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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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Anna Jenčová. 2017-06-30. R\'enyi relative entropies and noncommutative $L_p$-spaces II. https://doi.org/10.1007/s00023-021-01074-9

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