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Title: Sandwiched Rényi divergence satisfies data processing inequality

Abstract

Sandwiched (quantum) α-Rényi divergence has been recently defined in the independent works of Wilde et al. [“Strong converse for the classical capacity of entanglement-breaking channels,” preprint http://arxiv.org/abs/arXiv:1306.1586 (2013)] and Müller-Lennert et al. [“On quantum Rényi entropies: a new definition, some properties and several conjectures,” preprint http://arxiv.org/abs/arXiv:1306.3142v1 (2013)]. This new quantum divergence has already found applications in quantum information theory. Here we further investigate properties of this new quantum divergence. In particular, we show that sandwiched α-Rényi divergence satisfies the data processing inequality for all values of α > 1. Moreover we prove that α-Holevo information, a variant of Holevo information defined in terms of sandwiched α-Rényi divergence, is super-additive. Our results are based on Hölder's inequality, the Riesz-Thorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.

Authors:
 [1]
  1. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran (Iran, Islamic Republic of)
Publication Date:
OSTI Identifier:
22251255
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 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DATA PROCESSING; ENTROPY; INTERPOLATION; QUANTUM ENTANGLEMENT; QUANTUM INFORMATION

Citation Formats

Beigi, Salman. Sandwiched Rényi divergence satisfies data processing inequality. United States: N. p., 2013. Web. doi:10.1063/1.4838855.
Beigi, Salman. Sandwiched Rényi divergence satisfies data processing inequality. United States. https://doi.org/10.1063/1.4838855
Beigi, Salman. 2013. "Sandwiched Rényi divergence satisfies data processing inequality". United States. https://doi.org/10.1063/1.4838855.
@article{osti_22251255,
title = {Sandwiched Rényi divergence satisfies data processing inequality},
author = {Beigi, Salman},
abstractNote = {Sandwiched (quantum) α-Rényi divergence has been recently defined in the independent works of Wilde et al. [“Strong converse for the classical capacity of entanglement-breaking channels,” preprint http://arxiv.org/abs/arXiv:1306.1586 (2013)] and Müller-Lennert et al. [“On quantum Rényi entropies: a new definition, some properties and several conjectures,” preprint http://arxiv.org/abs/arXiv:1306.3142v1 (2013)]. This new quantum divergence has already found applications in quantum information theory. Here we further investigate properties of this new quantum divergence. In particular, we show that sandwiched α-Rényi divergence satisfies the data processing inequality for all values of α > 1. Moreover we prove that α-Holevo information, a variant of Holevo information defined in terms of sandwiched α-Rényi divergence, is super-additive. Our results are based on Hölder's inequality, the Riesz-Thorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.},
doi = {10.1063/1.4838855},
url = {https://www.osti.gov/biblio/22251255}, journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 12,
volume = 54,
place = {United States},
year = {Sun Dec 15 00:00:00 EST 2013},
month = {Sun Dec 15 00:00:00 EST 2013}
}