Research PaperNo Access

A strengthened data processing inequality for the Belavkin–Staszewski relative entropy

Andreas Bluhm

http://orcid.org/0000-0003-4796-7633

Zentrum Mathematik, Technische Universität München, Boltzmannstrasse 3, Garching, 85748, Germany

E-mail Address: andreas.bluhm@ma.tum.de

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and

Ángela Capel

http://orcid.org/0000-0001-6713-6760

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13–15, Madrid, 28049, Spain

E-mail Address: angela.capel@icmat.es

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https://doi.org/10.1142/S0129055X20500051Cited by:11 (Source: Crossref)

Abstract

In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard $f$ -divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal $f$ -divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.

Keywords:

AMSC: 81P45, 81P16, 94A17

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