Communications in Mathematical Physics

, Volume 328, Issue 1, pp 251–284 | Cite as

One-Shot Decoupling

  • Frédéric Dupuis
  • Mario Berta
  • Jürg Wullschleger
  • Renato Renner
Open Access


If a quantum system A, which is initially correlated to another system, E, undergoes an evolution separated from E, then the correlation to E generally decreases. Here, we study the conditions under which the correlation disappears (almost) completely, resulting in a decoupling of A from E. We give a criterion for decoupling in terms of two smooth entropies, one quantifying the amount of initial correlation between A and E, and the other characterizing the mapping that describes the evolution of A. The criterion applies to arbitrary such mappings in the general one-shot setting. Furthermore, the criterion is tight for mappings that satisfy certain natural conditions. One-shot decoupling has a number of applications both in physics and information theory, e.g., as a building block for quantum information processing protocols. As an example, we give a one-shot state merging protocol and show that it is essentially optimal in terms of its entanglement consumption/production.


Partial Trace Trace Distance Local Isometry Noisy Quantum Channel Asymptotic Equipartition Property 
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  1. Abe13.
    Aberg J.: Truly work-like work extraction via a single-shot analysis. Nat. Commun. 4, 1925 (2013)ADSCrossRefGoogle Scholar
  2. ADHW09.
    Abeyesinghe A., Devetak I., Hayden P., Winter A.: The mother of all protocols: restructuring quantum information’s family tree. Proc. Roy. Soc. A 465, 2537 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. BBCM95.
    Bennett C.H., Brassard G., Crépeau C., Maurer U.: Generalized privacy amplification. IEEE Trans. Info. Theory 41, 1915 (1995)CrossRefzbMATHGoogle Scholar
  4. BCR11.
    Berta M., Christandl M., Renner R.: The quantum reverse Shannon theorem based on one-shot information theory. Commun. Math. Phys. 306, 579 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. BDH+09.
    Bennett Charles, H., Devetak, I., Harrow, A.W., Shor, P.W., Winter, A.: Quantum reverse Shannon theorem. (2009). arXiv:0912.5537v2Google Scholar
  6. Ber08.
    Berta, M.: Single-shot quantum state merging. Diploma Thesis, ETH Zurich, (2008). arXiv:0912.4495v1Google Scholar
  7. BH13.
    Brandao, F.G.S.L., Horodecki, M.: An area law for entanglement from exponential decay of correlations. Nat. Phys. 9, 721 (2013)Google Scholar
  8. BP07.
    Braunstein S.L., Pati A.K.: Quantum information cannot be completely hidden in correlations: implications for the black-hole information paradox. Phys. Rev. Lett. 98, 080502 (2007)ADSCrossRefMathSciNetGoogle Scholar
  9. BRW07.
    Berta, M., Renner, R., Winter, A.: Tightness of decoupling by projective measurements. Unpublished manuscript; the technical proof appeared as part of [Ber08] (2007)Google Scholar
  10. BSST02.
    Bennett C.H., Shor P.W., Smolin J.A., Thapliyal A.V.: Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Info. Theory 48, 2637 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  11. Bus09.
    Buscemi F.: Private quantum decoupling and secure disposal of information. New J. Phys. 11, 123002 (2009)ADSCrossRefMathSciNetGoogle Scholar
  12. Cho75.
    Choi M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285 (1975)CrossRefzbMATHGoogle Scholar
  13. CS06.
    Collins B., Śniady P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773 (2006)ADSCrossRefzbMATHGoogle Scholar
  14. Dat09.
    Datta N.: Min- and max- relative entropies and a new entanglement monotone. IEEE Trans. Info. Theory 55, 2816 (2009)CrossRefGoogle Scholar
  15. Dev05.
    Devetak I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Info. Theory 51, 44 (2005)CrossRefMathSciNetGoogle Scholar
  16. dRAR+11.
    Lídia, del R., Åberg, J., Renner, R., Dahlsten, O., Vedral, V. The thermodynamic meaning of negative entropy. Nature, 474, 61 (2011)Google Scholar
  17. DRRV09.
    Dahlsten O., Renner R., Rieper E., Vedral V.: Inadequacy of von Neumann entropy for characterizing extractable work. New J. Phys. 13, 053015 (2009)CrossRefGoogle Scholar
  18. Dup09.
    Dupuis, F.: The decoupling approach to quantum information theory. PhD thesis, Université de Montréal, (2009). arXiv:1004.1641v1Google Scholar
  19. FDOR12.
    Faist, P., Dupuis, F., Oppenheim, J., Renner, R.: A quantitative Landauer’s principle (2012). arXiv:1211.1037v1Google Scholar
  20. GPW05.
    Groisman B., Popescu S., Winter A.: Quantum, classical, and total amount of correlations in quantum state. Phys. Rev. A 72, 032317 (2005)ADSCrossRefMathSciNetGoogle Scholar
  21. HHWY08.
    Hayden P., Horodecki M., Winter A., Yard J.: A decoupling approach to the quantum capacity. Open Syst. Info. Dynam. 15, 7 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  22. HO13.
    Horodecki M., Oppenheim J.: Fundament limitations for quantum and nanoscale thermodynamics. Nat. Commun. 4, 2059 (2013)ADSCrossRefGoogle Scholar
  23. HOW05.
    Horodecki, M., Oppenheim, J., Winter, A.: Partial quantum information. Nature, 436, 673 (2005)Google Scholar
  24. HOW07.
    Horodecki M., Oppenheim J., Winter A.: Quantum state merging and negative information. Commun. Math. Phys. 269, 107 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. HP07.
    Hayden P., Preskill J.: Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys. 07, 120 (2007)CrossRefMathSciNetGoogle Scholar
  26. Hut11.
    Hutter, A.: Understanding thermalization from decoupling. Master Thesis, ETH Zurich, (2011).
  27. Jam72.
    Jamiołkowski A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Reports Math. Phys. 3, 275 (1972)ADSCrossRefzbMATHGoogle Scholar
  28. KRS09.
    König R., Renner R., Schaffner C.: The operational meaning of min- and max-entropy. IEEE Trans. Info. Theory 55, 4337 (2009)CrossRefGoogle Scholar
  29. Llo97.
    Lloyd S.: Capacity of the noisy quantum channel. Phys. Rev. A 55, 1613 (1997)ADSCrossRefMathSciNetGoogle Scholar
  30. LPSW09.
    Linden N., Popescu S., Short A.J., Winter A.: Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009)ADSCrossRefMathSciNetGoogle Scholar
  31. Par89a.
    Partovi M.H.: Irreversibility, reduction, and entropy increase in quantum measurements. Phys. Lett. A 137, 445 (1989)ADSCrossRefMathSciNetGoogle Scholar
  32. Par89b.
    Partovi M.H.: Quantum thermodynamics. Phys. Lett. A 137, 440 (1989)ADSCrossRefMathSciNetGoogle Scholar
  33. PZ13.
    Braunstein Stefano Pirandola S.L., Zyczkowski K.: Better late than never: Information retrieval from black holes. Phys. Rev. Lett. 110, 101301 (2013)ADSCrossRefGoogle Scholar
  34. Ren05.
    Renner, R.: Security of quantum key distribution. PhD thesis, ETH Zurich, (2005).
  35. Ren09.
    Renner, R.: Optimal decoupling. Proc. Intern. Congr. Math. Phys, p. 541, (2009)Google Scholar
  36. RK05.
    Renner, R., Robert, K.: Universally composable privacy amplification against quantum adversaries. In: Second Theory of Cryptography Conference TCC, vol. 3378 of Lecture Notes in Computer Science, p. 407. Springer, (2005)Google Scholar
  37. RW04.
    Renner, R., Stefan, W.: Smooth Rényi entropy and applications. In: Proceedings International Symposium on Information Theory, p. 233, (2004)Google Scholar
  38. Sho02.
    Peter, S.: The quantum channel capacity and coherent information. Lecture notes, MSRI workshop on quantum computation, (2002).
  39. Sti55.
    Stinespring W.F.: Positive function on C*-algebras. Proc. Amer. Math. Soc. 6, 211 (1955)zbMATHMathSciNetGoogle Scholar
  40. TCR09.
    Marco T., Roger C., Renner R.: A fully quantum asymptotic equipartition property. IEEE Trans. Info. Theory 55, 5840 (2009)CrossRefGoogle Scholar
  41. TCR10.
    Marco T., Roger C., Renner R.: Duality between smooth min- and max-entropies. IEEE Trans. Info. Theory 56, 4674 (2010)CrossRefGoogle Scholar
  42. Tom12.
    Marco, T.: A framework for non-asymptotic quantum information theory. PhD thesis, ETH Zurich, (2012). arXiv:1203.2142v2Google Scholar
  43. TRSS10.
    Marco, T., Renner, R., Christian, S., Adam, S.: Leftover hashing against quantum side information. In: Information Theory Proceedings (ISIT), 2010 IEEE International Symposium on, p. 2703, (2010)Google Scholar
  44. Uhl76.
    Uhlmann A.: The ‘transition probability’ in the state space of a *-algebra. Reports Math. Phys. 9, 273 (1976)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  45. Wat08.
    John, W.: Theory of quantum information—Lecture notes from fall 2008. (2008).

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© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • Frédéric Dupuis
    • 1
  • Mario Berta
    • 1
  • Jürg Wullschleger
    • 2
    • 3
  • Renato Renner
    • 1
  1. 1.Institute for Theoretical PhysicsETH ZurichZurichSwitzerland
  2. 2.Department of Computer Science and Operations ResearchUniversité de MontréalMontrealCanada
  3. 3.McGill UniversityMontrealCanada

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