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.
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Acknowledgements
I am indebted to Fumio Hiai for sharing the manuscript of his monograph [10] and useful discussions and comments. His variational formula and its proof inspired a large part of this paper. The research was supported by the Grants APVV-16-0073 and VEGA 2/0142/20.
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Communicated by Matthias Christandl.
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Jenčová, A. Rényi Relative Entropies and Noncommutative \(L_p\)-Spaces II. Ann. Henri Poincaré 22, 3235–3254 (2021). https://doi.org/10.1007/s00023-021-01074-9
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DOI: https://doi.org/10.1007/s00023-021-01074-9