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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
Article

Abstract

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.

Notes

Acknowledgements

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.

References

  1. 1.
    Audenaert, K.M.R.: A sharp continuity estimate for the von Neumann entropy. J. Phys. A Math. Theor. 40(28), 8127 (2007)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Barnum, H., Knill, E.: Reversing quantum dynamics with near-optimal quantum and classical fidelity. J. Math. Phys. 43(5), 2097–2106 (2002). arXiv:quant-ph/0004088 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54(5), 3824–3851 (1996)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Berta, M., Lemm, M., Wilde, M.M.: Monotonicity of quantum relative entropy and recoverability. Quantum Inf. Comput. 15(15 & 16), 1333–1354 (2015)MathSciNetGoogle Scholar
  5. 5.
    Bhatia, R.: Matrix Analysis. Springer, Berlin (1997)zbMATHCrossRefGoogle Scholar
  6. 6.
    Brandão, F.G.S.L., Harrow, A.W., Oppenheim, J., Strelchuk, S.: Quantum conditional mutual information, reconstructed states, and state redistribution. Phys. Rev. Lett. 115(5), 050501 (2015)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Devetak, I., Yard, J.: Exact cost of redistributing multipartite quantum states. Phys. Rev. Lett. 100(23), 230501 (2008)ADSCrossRefGoogle Scholar
  8. 8.
    Fawzi, O., Renner, R.: Quantum conditional mutual information and approximate Markov chains. Commun. Math. Phys. 340(2), 575–611 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Fuchs, C., van de Graaf, J.: Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inf. Theory 45(4), 1216–1227 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Grafakos, L.: Classical Fourier Analysis, 2nd edn. Springer, Berlin (2008)zbMATHGoogle Scholar
  11. 11.
    Grümm, H.: Two theorems about \({C}_p\). Rep. Math. Phys. 4(3), 211–215 (1973)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hiai, F., Mosonyi, M., Petz, D., Beny, C.: Quantum \(f\)-divergences and error correction. Rev. Math. Phys. 23(7), 691–747 (2011). arXiv:1008.2529 MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hirschman, I.I.: A convexity theorem for certain groups of transformations. Journal d’Analyse Mathématique 2(2), 209–218 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Holevo, A.S.: Quantum Systems, Channels, Information. De Gruyter Studies in Mathematical Physics 16, de Gruyter (2012)Google Scholar
  15. 15.
    Jencǒvà, A.: Randomization Theorems for Quantum Channels. arXiv:1404.3900 (2014)
  16. 16.
    Jencǒvà, A., Petz, D.: Sufficiency in quantum statistical inference. Commun. Math. Phys. 263(1), 259–276 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Jencǒvà, A., Petz, D.: Sufficiency in quantum statistical inference: a survey with examples. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 09(03), 331–351 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Knill, E., Laflamme, R.: Theory of quantum error-correcting codes. Phys. Rev. A 55(2), 900–911 (1997)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Li, K., Winter, A.: Squashed Entanglement, k-Extendibility, Quantum Markov Chains, and Recovery Maps. Found. Phys. (2018).  https://doi.org/10.1007/s10701-018-0143-6
  20. 20.
    Lieb, E.H., Ruskai, M.B.: A fundamental property of quantum-mechanical entropy. Phys. Rev. Lett. 30, 434–436 (1973)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14(12), 1938–1941 (1973)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Lindblad, G.: Entropy, information and quantum measurements. Commun. Math. Phys. 33, 305–322 (1973)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Lindblad, G.: Expectations and entropy inequalities for finite quantum systems. Commun. Math. Phys. 39(2), 111–119 (1974)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Lindblad, G.: Completely positive maps and entropy inequalities. Commun. Math. Phys. 40(2), 147–151 (1975)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Mandayam, P., Ng, H.K.: Towards a unified framework for approximate quantum error correction. Phys. Rev. A 86(1), 012335 (2012). arXiv:1202.5139 ADSCrossRefGoogle Scholar
  26. 26.
    Mosonyi, M., Petz, D.: Structure of sufficient quantum coarse-grainings. Lett. Math. Phys. 68(1), 19–30 (2004)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  28. 28.
    Ohya, M., Petz, D.: Quantum Entropy and Its Use. Springer, Berlin (1993)zbMATHCrossRefGoogle Scholar
  29. 29.
    Petz, D.: Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Commun. Math. Phys. 105(1), 123–131 (1986)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Petz, D.: Sufficiency of channels over von Neumann algebras. Q. J. Math. 39(1), 97–108 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Petz, D.: Monotonicity of quantum relative entropy revisited. Rev. Math. Phys. 15(01), 79–91 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Powers, R.T., Størmer, E.: Free states of the canonical anticommutation relations. Commun. Math. Phys. 16(1), 1–33 (1970)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Rudin, W.: Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics, 3rd edn. McGraw-Hill Book Co, New York (1976)zbMATHGoogle Scholar
  34. 34.
    Ruskai, M.B.: Inequalities for quantum entropy: a review with conditions for equality. J. Math. Phys. 43(9), 4358–4375 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Seshadreesan, K.P., Berta, M., Wilde, M.M.: Rényi squashed entanglement, discord, and relative entropy differences. J. Phys. A Math. Theor. 48(39), 395303 (2015)zbMATHCrossRefGoogle Scholar
  36. 36.
    Shirokov, M.E.: Measures of quantum correlations in infinite-dimensional systems. Sbornik Math. 207(5), 724 (2015). arXiv:1506.06377 ADSzbMATHCrossRefGoogle Scholar
  37. 37.
    Simon, B.: Trace Ideals and Their Applications, 2nd edn. American Mathematical Society, Providence (2005)zbMATHGoogle Scholar
  38. 38.
    Sutter, D., Fawzi, O., Renner, R.: Universal recovery map for approximate Markov chains. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 472, 2186 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Sutter, D., Tomamichel, M., Harrow, A.W.: Strengthened monotonicity of relative entropy via pinched Petz recovery map. IEEE Trans. Inf. Theory 62(5), 2907–2913 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Uhlmann, A.: The “transition probability” in the state space of a *-algebra. Rep. Math. Phys. 9(2), 273–279 (1976)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Uhlmann, A.: Relative entropy and the Wigner–Yanase–Dyson–Lieb concavity in an interpolation theory. Commun. Math. Phys. 54(1), 21–32 (1977)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Wilde, M.M.: Recoverability in quantum information theory. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 471(2182), 20150338 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Winter, A.: Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints. Commun. Math. Phys. 347(1), 291–313 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Winter, A., Li, K.: A Stronger Subadditivity Relation? With Applications to Squashed Entanglement, Sharability and Separability. http://www.scribd.com/document/337859204 (2012)

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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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