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On Variational Expressions for Quantum Relative Entropies

Berta, Mario and Fawzi, Omar and Tomamichel, Marco (2017) On Variational Expressions for Quantum Relative Entropies. Letters in Mathematical Physics, 107 (12). pp. 2239-2265. ISSN 0377-9017. doi:10.1007/s11005-017-0990-7.

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Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback–Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki’s quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz’ conclusion remains true if we allow general positive operator-valued measures. Second, we extend the result to Rényi relative entropies and show that for non-commuting states the sandwiched Rényi relative entropy is strictly larger than the measured Rényi relative entropy for α∈(1/2,∞) and strictly smaller for α∈[0,1/2). The latter statement provides counterexamples for the data processing inequality of the sandwiched Rényi relative entropy for α<1/2. Our main tool is a new variational expression for the measured Rényi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive.

Item Type:Article
Related URLs:
URLURL TypeDescription ReadCube access Paper
Berta, Mario0000-0002-0428-3429
Tomamichel, Marco0000-0001-5410-3329
Additional Information:© 2017 Springer Science+Business Media B.V. Received: 14 February 2017; Revised: 22 August 2017; Accepted: 23 August 2017; Published online: 7 September 2017. We acknowledge discussions with Fernando Brandão, Douglas Farenick and Hamza Fawzi. MB acknowledges funding by the SNSF through a fellowship, funding by the Institute for Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (NFS Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028), and funding support form the ARO grant for Research on Quantum Algorithms at the IQIM (W911NF-12-1-0521). Most of this work was done while OF was also with the Department of Computing and Mathematical Sciences, California Institute of Technology. MT would like to thank the IQIM at CalTech and John Preskill for his hospitality during the time most of the technical aspects of this project were completed. He is funded by an ARC Discovery Early Career Researcher Award fellowship (Grant No. DE160100821).
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Swiss National Science Foundation (SNSF)UNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Gordon and Betty Moore FoundationGBMF-12500028
Army Research Office (ARO)W911NF-12-1-0521
Australian Research CouncilDE160100821
Subject Keywords:Quantum entropy; Measured relative entropy; Relative entropy of recovery; Additivity in quantum information theory; Operator Jensen inequality; Convex optimization
Issue or Number:12
Classification Code:MSC: 94A17, 81Q99, 15A45
Record Number:CaltechAUTHORS:20161024-110728241
Persistent URL:
Official Citation:Berta, M., Fawzi, O. & Tomamichel, M. Lett Math Phys (2017) 107: 2239.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:71383
Deposited By: Melissa Ray
Deposited On:24 Oct 2016 19:13
Last Modified:11 Nov 2021 04:44

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