Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aberg J.: Truly work-like work extraction via a single-shot analysis. Nat. Commun. 4, 1925 (2013)
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)
Bennett C.H., Brassard G., Crépeau C., Maurer U.: Generalized privacy amplification. IEEE Trans. Info. Theory 41, 1915 (1995)
Berta M., Christandl M., Renner R.: The quantum reverse Shannon theorem based on one-shot information theory. Commun. Math. Phys. 306, 579 (2011)
Bennett Charles, H., Devetak, I., Harrow, A.W., Shor, P.W., Winter, A.: Quantum reverse Shannon theorem. (2009). arXiv:0912.5537v2
Berta, M.: Single-shot quantum state merging. Diploma Thesis, ETH Zurich, (2008). arXiv:0912.4495v1
Brandao, F.G.S.L., Horodecki, M.: An area law for entanglement from exponential decay of correlations. Nat. Phys. 9, 721 (2013)
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)
Berta, M., Renner, R., Winter, A.: Tightness of decoupling by projective measurements. Unpublished manuscript; the technical proof appeared as part of [Ber08] (2007)
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)
Buscemi F.: Private quantum decoupling and secure disposal of information. New J. Phys. 11, 123002 (2009)
Choi M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285 (1975)
Collins B., Śniady P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773 (2006)
Datta N.: Min- and max- relative entropies and a new entanglement monotone. IEEE Trans. Info. Theory 55, 2816 (2009)
Devetak I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Info. Theory 51, 44 (2005)
Lídia, del R., Åberg, J., Renner, R., Dahlsten, O., Vedral, V. The thermodynamic meaning of negative entropy. Nature, 474, 61 (2011)
Dahlsten O., Renner R., Rieper E., Vedral V.: Inadequacy of von Neumann entropy for characterizing extractable work. New J. Phys. 13, 053015 (2009)
Dupuis, F.: The decoupling approach to quantum information theory. PhD thesis, Université de Montréal, (2009). arXiv:1004.1641v1
Faist, P., Dupuis, F., Oppenheim, J., Renner, R.: A quantitative Landauer’s principle (2012). arXiv:1211.1037v1
Groisman B., Popescu S., Winter A.: Quantum, classical, and total amount of correlations in quantum state. Phys. Rev. A 72, 032317 (2005)
Hayden P., Horodecki M., Winter A., Yard J.: A decoupling approach to the quantum capacity. Open Syst. Info. Dynam. 15, 7 (2008)
Horodecki M., Oppenheim J.: Fundament limitations for quantum and nanoscale thermodynamics. Nat. Commun. 4, 2059 (2013)
Horodecki, M., Oppenheim, J., Winter, A.: Partial quantum information. Nature, 436, 673 (2005)
Horodecki M., Oppenheim J., Winter A.: Quantum state merging and negative information. Commun. Math. Phys. 269, 107 (2007)
Hayden P., Preskill J.: Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys. 07, 120 (2007)
Hutter, A.: Understanding thermalization from decoupling. Master Thesis, ETH Zurich, (2011). http://www.quantumlah.org/media/thesis/NCQT_AdrianHutter_MSc2011.pdf
Jamiołkowski A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Reports Math. Phys. 3, 275 (1972)
König R., Renner R., Schaffner C.: The operational meaning of min- and max-entropy. IEEE Trans. Info. Theory 55, 4337 (2009)
Lloyd S.: Capacity of the noisy quantum channel. Phys. Rev. A 55, 1613 (1997)
Linden N., Popescu S., Short A.J., Winter A.: Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009)
Partovi M.H.: Irreversibility, reduction, and entropy increase in quantum measurements. Phys. Lett. A 137, 445 (1989)
Partovi M.H.: Quantum thermodynamics. Phys. Lett. A 137, 440 (1989)
Braunstein Stefano Pirandola S.L., Zyczkowski K.: Better late than never: Information retrieval from black holes. Phys. Rev. Lett. 110, 101301 (2013)
Renner, R.: Security of quantum key distribution. PhD thesis, ETH Zurich, (2005). http://www.worldscientific.com/doi/abs/10.1142/S0219749908003256
Renner, R.: Optimal decoupling. Proc. Intern. Congr. Math. Phys, p. 541, (2009)
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)
Renner, R., Stefan, W.: Smooth Rényi entropy and applications. In: Proceedings International Symposium on Information Theory, p. 233, (2004)
Peter, S.: The quantum channel capacity and coherent information. Lecture notes, MSRI workshop on quantum computation, (2002). http://www.msri.org/publications/ln/msri/2002/quantumcrypto/shor/1/
Stinespring W.F.: Positive function on C*-algebras. Proc. Amer. Math. Soc. 6, 211 (1955)
Marco T., Roger C., Renner R.: A fully quantum asymptotic equipartition property. IEEE Trans. Info. Theory 55, 5840 (2009)
Marco T., Roger C., Renner R.: Duality between smooth min- and max-entropies. IEEE Trans. Info. Theory 56, 4674 (2010)
Marco, T.: A framework for non-asymptotic quantum information theory. PhD thesis, ETH Zurich, (2012). arXiv:1203.2142v2
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)
Uhlmann A.: The ‘transition probability’ in the state space of a *-algebra. Reports Math. Phys. 9, 273 (1976)
John, W.: Theory of quantum information—Lecture notes from fall 2008. (2008). http://www.cs.uwaterloo.ca/~watrous/quant-info/
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. B. Ruskai
Rights and permissions
Open Access This 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.
About this article
Cite this article
Dupuis, F., Berta, M., Wullschleger, J. et al. One-Shot Decoupling. Commun. Math. Phys. 328, 251–284 (2014). https://doi.org/10.1007/s00220-014-1990-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-1990-4