On quantum Rényi entropies: A new generalization and some properties
Abstract
The Rényi entropies constitute a family of information measures that generalizes the wellknown Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasientropies and Renner's conditional min, max, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, minentropy, collision entropy, and the maxentropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including dataprocessing inequalities, a duality relation, and an entropic uncertainty relation.
 Authors:

 Department of Mathematics, ETH Zurich, 8092 Zürich (Switzerland)
 Department of Computer Science, Aarhus University, 8200 Aarhus (Denmark)
 Department of Mathematics, Technische Universität München, 85748 Garching (Germany)
 CWI (Centrum Wiskunde and Informatica), 1090 Amsterdam (Netherlands)
 Centre for Quantum Technologies, National University of Singapore, Singapore 117543 (Singapore)
 Publication Date:
 OSTI Identifier:
 22251256
 Resource Type:
 Journal Article
 Journal Name:
 Journal of Mathematical Physics
 Additional Journal Information:
 Journal Volume: 54; Journal Issue: 12; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 00222488
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DATA PROCESSING; ENTROPY; INFORMATION THEORY
Citation Formats
MüllerLennert, Martin, Dupuis, Frédéric, Szehr, Oleg, Fehr, Serge, and Tomamichel, Marco. On quantum Rényi entropies: A new generalization and some properties. United States: N. p., 2013.
Web. doi:10.1063/1.4838856.
MüllerLennert, Martin, Dupuis, Frédéric, Szehr, Oleg, Fehr, Serge, & Tomamichel, Marco. On quantum Rényi entropies: A new generalization and some properties. United States. doi:10.1063/1.4838856.
MüllerLennert, Martin, Dupuis, Frédéric, Szehr, Oleg, Fehr, Serge, and Tomamichel, Marco. Sun .
"On quantum Rényi entropies: A new generalization and some properties". United States. doi:10.1063/1.4838856.
@article{osti_22251256,
title = {On quantum Rényi entropies: A new generalization and some properties},
author = {MüllerLennert, Martin and Dupuis, Frédéric and Szehr, Oleg and Fehr, Serge and Tomamichel, Marco},
abstractNote = {The Rényi entropies constitute a family of information measures that generalizes the wellknown Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasientropies and Renner's conditional min, max, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, minentropy, collision entropy, and the maxentropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including dataprocessing inequalities, a duality relation, and an entropic uncertainty relation.},
doi = {10.1063/1.4838856},
journal = {Journal of Mathematical Physics},
issn = {00222488},
number = 12,
volume = 54,
place = {United States},
year = {2013},
month = {12}
}