Annales Henri Poincaré

, Volume 19, Issue 10, pp 2955–2978 | Cite as

Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy

  • Marius Junge
  • Renato Renner
  • David Sutter
  • Mark M. Wilde
  • Andreas Winter
Open Access


The data processing inequality states that the quantum relative entropy between two states \(\rho \) and \(\sigma \) can never increase by applying the same quantum channel \(\mathcal {N}\) to both states. This inequality can be strengthened with a remainder term in the form of a distance between \(\rho \) and the closest recovered state \((\mathcal {R} \circ \mathcal {N})(\rho )\), where \(\mathcal {R}\) is a recovery map with the property that \(\sigma = (\mathcal {R} \circ \mathcal {N})(\sigma )\). We show the existence of an explicit recovery map that is universal in the sense that it depends only on \(\sigma \) and the quantum channel \(\mathcal {N}\) to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.



We thank Omar Fawzi, Rupert Frank, Jürg Fröhlich, Elliott Lieb, Volkher Scholz and Marco Tomamichel for helpful discussions. We further thank the anonymous referee for constructive feedback. MJ’s work was supported by NSF DMS 1501103 and NSF-DMS 1201886. RR and DS acknowledge support by the European Research Council (ERC) via grant No. 258932, 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.” MMW acknowledges support from the NSF under Award Nos. CCF-1350397 & 1714215 and the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019. He is also grateful to co-authors MJ, RR, and AW for hospitality and support during research visits in summer 2015. AWs work was supported by the EU (STREP RAQUEL), the ERC (AdG IRQUAT), the Spanish MINECO (grant FIS2013-40627-P) with the support of FEDER funds, as well as by the Generalitat de Catalunya CIRIT, project 2014-SGR-966.


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Open AccessThis 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.

Authors and Affiliations

  • Marius Junge
    • 1
  • Renato Renner
    • 2
  • David Sutter
    • 2
  • Mark M. Wilde
    • 3
  • Andreas Winter
    • 4
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Institute for Theoretical PhysicsETH ZurichZurichSwitzerland
  3. 3.Department of Physics and Astronomy, Center for Computation and Technology, Hearne Institute for Theoretical PhysicsLouisiana State UniversityBaton RougeUSA
  4. 4.ICREA & Física Teòrica: Informació i Fenòmens QuànticsUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain

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