Rényi Relative Entropies and Noncommutative Lp-Spaces
Abstract
We propose an extension of the sandwiched Rényi relative α-entropy to normal positive functionals on arbitrary von Neumann algebras, for the values α>1. For this, we use Kosaki's definition of noncommutative Lp-spaces with respect to a state. We show that these extensions coincide with the previously defined Araki-Masuda divergences (Berta et al. in arXiv:1608.05317, 2016) and prove some of their properties, in particular the data processing inequality with respect to positive normal unital maps. As a consequence, we obtain monotonicity of the Araki relative entropy with respect to such maps, extending the results of Müller-Hermes and Reeb. (Ann. Henri Poincaré 18:1777-1788, 2017) to arbitrary von Neumann algebras. It is also shown that equality in data processing inequality characterizes sufficiency (reversibility) of quantum channels.
- Publication:
-
Annales Henri Poincaré
- Pub Date:
- August 2018
- DOI:
- 10.1007/s00023-018-0683-5
- arXiv:
- arXiv:1609.08462
- Bibcode:
- 2018AnHP...19.2513J
- Keywords:
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- Quantum Physics;
- Mathematical Physics;
- Mathematics - Operator Algebras
- E-Print:
- Accepted manuscript