Berta, Mario and Furrer, Fabian and Scholz, Volkher B. (2016) The smooth entropy formalism for von Neumann algebras. Journal of Mathematical Physics, 57 (1). Art. No. 015213. ISSN 0022-2488. doi:10.1063/1.4936405. https://resolver.caltech.edu/CaltechAUTHORS:20160225-124944101
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Abstract
We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy, we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.
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| Additional Information: | © 2016 AIP Publishing LLC. Received 14 August 2015; accepted 12 November 2015; published online 1 December 2015. We thank Renato Renner for instructive discussions about privacy amplification. We would also like to thank Marco Tomamichel for many insightful discussions about the smooth entropy formalism, and for detailed feedback on the first version of this paper. We acknowledge discussions with Matthias Christandl, Reinhard F. Werner, Michael Walter, and Joseph M. Renes. We thank an anonymous reviewer for pointing out an error in the proof of Lemma 20 and a detailed explanation of how to fix it. M.B. and V.B.S. are both grateful for the hospitality and the inspiring working environment at the Institute Mittag-Leffler in Djursholm, Sweden, where this work was started. Most of this work was done while M.B. was at ETH Zurich, and F.F. and V.B.S. were at the University of Hanover. M.B. acknowledges funding provided by the Institute for Quantum Information and Matter, a NSF Physics Frontiers Center (NFS Grant No. PHY-1125565) with support of the Gordon and Betty Moore Foundation (No. GBMF-12500028). Additional funding support was provided by the ARO grant for Research on Quantum Algorithms at the IQIM (No. W911NF-12-1-0521). F.F. acknowledges support from the Graduiertenkolleg 1463 of the Leibniz University Hanover and by the Japan Society for the Promotion of Science (JSPS) by KAKENHI Grant No. 24-02793, and F.F. and V.B.S. both acknowledge support by the BMBF project QUOREP as well as the DFG cluster of excellence QUEST. | ||||||||||||||||||
| Group: | Institute for Quantum Information and Matter | ||||||||||||||||||
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| Issue or Number: | 1 | ||||||||||||||||||
| DOI: | 10.1063/1.4936405 | ||||||||||||||||||
| Record Number: | CaltechAUTHORS:20160225-124944101 | ||||||||||||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20160225-124944101 | ||||||||||||||||||
| Official Citation: | The smooth entropy formalism for von Neumann algebras Mario Berta, Fabian Furrer and Volkher B. Scholz J. Math. Phys. 57, 015213 (2016); http://dx.doi.org/10.1063/1.4936405 | ||||||||||||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||||||||
| ID Code: | 64760 | ||||||||||||||||||
| Collection: | CaltechAUTHORS | ||||||||||||||||||
| Deposited By: | Ruth Sustaita | ||||||||||||||||||
| Deposited On: | 25 Feb 2016 21:16 | ||||||||||||||||||
| Last Modified: | 10 Nov 2021 23:35 |
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