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Multivariate Trace Inequalities

Sutter, David and Berta, Mario and Tomamichel, Marco (2017) Multivariate Trace Inequalities. Communications in Mathematical Physics, 352 (1). pp. 37-58. ISSN 0010-3616. doi:10.1007/s00220-016-2778-5.

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We prove several trace inequalities that extend the Golden–Thompson and the Araki–Lieb–Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb’s triple matrix inequality. As an example application of our four matrix extension of the Golden–Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities.

Item Type:Article
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URLURL TypeDescription Paper ReadCube access
Berta, Mario0000-0002-0428-3429
Tomamichel, Marco0000-0001-5410-3329
Additional Information:© 2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Received: 30 April 2016; Accepted: 11 August 2016; First Online: 18 October 2016. Elliott H. Lieb’s talk at the “Beyond I.I.D. in Information Theory” workshop in Banff inspired us to derive an alternative proof for his triple matrix inequality. We thank Christian Majenz for allowing us to include his counterexample for a three matrix GT inequality without rotations. We would also like to thank Jürg Fröhlich, Aram W. Harrow, Spyridon Michalakis, and Renato Renner for useful discussions about trace inequalities. We thank Marius C. Lemm, Thomas Vidick, and Mark M. Wilde for comments on an earlier draft. MB and MT thank the Institute for Theoretical Physics at ETH Zurich for hosting them when this project was initiated. DS acknowledges support by the Swiss National Science Foundation (SNSF) via the National Centre of Competence in Research “QSIT” and by the European Commission via the project “RAQUEL”. 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 from the ARO grant for Research on Quantum Algorithms at the IQIM (W911NF-12-1-0521). MT is funded by an ARC Discovery Early Career Researcher Award (DECRA) fellowship and acknowledges support from the ARC Centre of Excellence for Engineered Quantum Systems (EQUS).
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Swiss National Science Foundation (SNSF)UNSPECIFIED
European CommissionUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Gordon and Betty Moore FoundationGBMF-12500028
Army Research Office (ARO)W911NF-12-1-0521
Australian Research CouncilUNSPECIFIED
Issue or Number:1
Record Number:CaltechAUTHORS:20160622-115009117
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Official Citation:Sutter, D., Berta, M. & Tomamichel, M. Commun. Math. Phys. (2017) 352: 37. doi:10.1007/s00220-016-2778-5
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:68588
Deposited By: Jacquelyn O'Sullivan
Deposited On:27 Jun 2016 17:29
Last Modified:11 Nov 2021 04:02

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