Chain rules for quantum Rényi entropies
Abstract
We present chain rules for a new definition of the quantum Rényi conditional entropy sometimes called the “sandwiched” Rényi conditional entropy. More precisely, we prove analogues of the equation H(AB|C) = H(A|BC) + H(B|C), which holds as an identity for the von Neumann conditional entropy. In the case of the Rényi entropy, this relation no longer holds as an equality but survives as an inequality of the form H{sub α}(AB|C) ⩾ H{sub β}(A|BC) + H{sub γ}(B|C), where the parameters α, β, γ obey the relation (α)/(α−1) =(β)/(β−1) +(γ)/(γ−1) and (α − 1)(β − 1)(γ − 1) > 1; if (α − 1)(β − 1)(γ − 1) < 1, the direction of the inequality is reversed.
- Authors:
-
- Faculty of Informatics, Masaryk University, Brno (Czech Republic)
- Publication Date:
- OSTI Identifier:
- 22403102
- Resource Type:
- Journal Article
- Journal Name:
- Journal of Mathematical Physics
- Additional Journal Information:
- Journal Volume: 56; Journal Issue: 2; Other Information: (c) 2015 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; ENTROPY; EQUATIONS; INFORMATION THEORY; QUANTUM MECHANICS
Citation Formats
Dupuis, Frédéric. Chain rules for quantum Rényi entropies. United States: N. p., 2015.
Web. doi:10.1063/1.4907981.
Dupuis, Frédéric. Chain rules for quantum Rényi entropies. United States. https://doi.org/10.1063/1.4907981
Dupuis, Frédéric. 2015.
"Chain rules for quantum Rényi entropies". United States. https://doi.org/10.1063/1.4907981.
@article{osti_22403102,
title = {Chain rules for quantum Rényi entropies},
author = {Dupuis, Frédéric},
abstractNote = {We present chain rules for a new definition of the quantum Rényi conditional entropy sometimes called the “sandwiched” Rényi conditional entropy. More precisely, we prove analogues of the equation H(AB|C) = H(A|BC) + H(B|C), which holds as an identity for the von Neumann conditional entropy. In the case of the Rényi entropy, this relation no longer holds as an equality but survives as an inequality of the form H{sub α}(AB|C) ⩾ H{sub β}(A|BC) + H{sub γ}(B|C), where the parameters α, β, γ obey the relation (α)/(α−1) =(β)/(β−1) +(γ)/(γ−1) and (α − 1)(β − 1)(γ − 1) > 1; if (α − 1)(β − 1)(γ − 1) < 1, the direction of the inequality is reversed.},
doi = {10.1063/1.4907981},
url = {https://www.osti.gov/biblio/22403102},
journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 2,
volume = 56,
place = {United States},
year = {Sun Feb 15 00:00:00 EST 2015},
month = {Sun Feb 15 00:00:00 EST 2015}
}
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