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Communications in Mathematical Physics

, Volume 340, Issue 2, pp 575–611 | Cite as

Quantum Conditional Mutual Information and Approximate Markov Chains

  • Omar FawziEmail author
  • Renato Renner
Open Access
Article

Abstract

A state on a tripartite quantum system \({A \otimes B \otimes C}\) forms a Markov chain if it can be reconstructed from its marginal on \({A \otimes B}\) by a quantum operation from B to \({B \otimes C}\). We show that the quantum conditional mutual information I(A : C|B) of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated.

Keywords

Density Operator Relative Entropy Trace Distance Conditional Mutual Information Strong Subadditivity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Alicki R., Fannes M.: Continuity of quantum conditional information. J. Phys. A Math. Gen. 37(5), L55–L57 (2004) arXiv:quant-ph/0312081 zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Beaudry N.J., Renner R.: An intuitive proof of the data processing inequality. Quantum Inf. Comput. 12(5–6), 432–441 (2012) arXiv:1107.0740 zbMATHMathSciNetGoogle Scholar
  3. 3.
    Berta, M., Christandl, M., Furrer, F., Scholz, V.B., Tomamichel, M.: Continuous variable entropic uncertainty relations in the presence of quantum memory (2013). arXiv:1308.4527
  4. 4.
    Berta M., Seshadreesan K., Wilde M.M.: Renyi generalizations of the conditional quantum mutual information. J. Math. Phys. 56, 022205 (2015) arXiv:1403.6102 MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Bhatia R.: Matrix Analysis. Springer, Berlin (1997)CrossRefGoogle 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, 050501 (2015) arXiv:1411.4921 CrossRefADSGoogle Scholar
  7. 7.
    Brandão F.G.S.L., Christandl M., Yard J.: Faithful squashed entanglement. Commun. Math. Phys. 306(3), 805–830 (2011) arXiv:1010.1750 zbMATHCrossRefADSGoogle Scholar
  8. 8.
    Brandão, F.G.S.L., Harrow, A.W.: Product-state approximations to quantum ground states. In: Proceedings of ACM STOC, pp. 871–880. ACM, New York (2013). arXiv:1310.0017
  9. 9.
    Brandão, F.G.S.L., Harrow, A.W.: Quantum de Finetti theorems under local measurements with applications. In: Proceedings of ACM STOC, pp. 861–870. ACM, New York (2013). arXiv:1210.6367
  10. 10.
    Braverman, M.: Interactive information complexity. In: Proceedings of ACM STOC, pp. 505–524. ACM, New York (2012). ECCC:TR11-123
  11. 11.
    Carlen E., Lieb E.: Remainder terms for some quantum entropy inequalities. J. Math. Phys. 55(4), 042201 (2014) arXiv:1402.3840 MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Choi M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10(3), 285–290 (1975)zbMATHCrossRefGoogle Scholar
  13. 13.
    Christandl, M.: The structure of bipartite quantum states—insights from group theory and cryptography. PhD thesis, University of Cambridge (2006). arXiv:quant-ph/0604183
  14. 14.
    Christandl M., König R., Mitchison G., Renner R.: One-and-a-half quantum de Finetti theorems. Commun. Math. Phys. 273(2), 473–498 (2007) arXiv:quant-ph/0602130 zbMATHCrossRefADSGoogle Scholar
  15. 15.
    Christandl M., König R., Renner R.: Postselection technique for quantum channels with applications to quantum cryptography. Phys. Rev. Lett. 102(2), 020504 (2009) arXiv:0809.3019 CrossRefADSGoogle Scholar
  16. 16.
    Christandl M., Schuch N., Winter A.: Entanglement of the antisymmetric state. Commun. Math. Phys. 311(2), 397–422 (2012) arXiv:0910.4151 zbMATHMathSciNetCrossRefADSGoogle Scholar
  17. 17.
    Christandl M., Winter A.: “Squashed entanglement”: an additive entanglement measure. J. Math. Phys. 45(3), 829–840 (2004)zbMATHMathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Cover T.M., Thomas J.A.: Elements of Information Theory. Wiley, New York (2005)CrossRefGoogle Scholar
  19. 19.
    Datta N.: Min-and max-relative entropies and a new entanglement monotone. IEEE Trans. Inform. Theory 55(6), 2816–2826 (2009) arXiv:0803.2770 MathSciNetCrossRefGoogle Scholar
  20. 20.
    Devetak I., Yard J.: Exact cost of redistributing multipartite quantum states. Phys. Rev. Lett. 100, 230501 (2008) arXiv:quant-ph/0612050 CrossRefADSGoogle Scholar
  21. 21.
    Dupuis, F., Krämer, L., Faist, P., Renes, J.M., Renner, R.: Generalized entropies. In: Proceedings of the XVIIth International Congress on Mathematical Physics, pp. 134–153 (2013). arXiv:1211.3141
  22. 22.
    Ekert, A., Renner, R.: The ultimate physical limits of privacy. Nature 507, 443–447 (2014)Google Scholar
  23. 23.
    Erker P.: How not to Rényi generalize the quantum conditional mutual information. J. Phys. A Math. Theor. 48, 275303 (2015) arXiv:1404.3628 CrossRefADSGoogle Scholar
  24. 24.
    Fuchs C.A., van de Graaf J.: Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inform. Theory 45(4), 1216–1227 (1999) arXiv:quant-ph/9712042 zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Furrer F., Åberg J., Renner R.: Min-and max-entropy in infinite dimensions. Commun. Math. Phys. 306(1), 165–186 (2011) arXiv:1004.1386 zbMATHCrossRefADSGoogle Scholar
  26. 26.
    Harrow, A.W.: Applications of coherent classical communication and the Schur transform to quantum information theory. PhD thesis, Massachusetts Institute of Technology (2005). arXiv:quant-ph/0512255
  27. 27.
    Hayashi M.: Quantum Information: An Introduction. Springer, Berlin (2006)Google Scholar
  28. 28.
    Hayden P., Jozsa R., Petz D., Winter A.: Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys. 246(2), 359–374 (2004) arXiv:quant-ph/0304007 zbMATHMathSciNetCrossRefADSGoogle Scholar
  29. 29.
    Hiai F., Petz D.: The proper formula for relative entropy and its asymptotics in quantum probability. Commun. Math. Phys. 143(1), 99–114 (1991)zbMATHMathSciNetCrossRefADSGoogle Scholar
  30. 30.
    Ibinson B., Linden N., Winter A.: Robustness of quantum Markov chains. Commun. Math. Phys. 277(2), 289–304 (2008) arXiv:quant-ph/0611057 zbMATHMathSciNetCrossRefADSGoogle Scholar
  31. 31.
    Jain, R., Nayak, A.: The space complexity of recognizing well-parenthesized expressions in the streaming model: the index function revisited. IEEE Trans. Inf. Theory 60(10) (2014). arXiv:1004.3165
  32. 32.
    Jain, R., Radhakrishnan, J., Sen, P.: A lower bound for the bounded round quantum communication complexity of set disjointness. In: Proceedings of FOCS, pp. 220–229. IEEE (2003). arXiv:quant-ph/0303138
  33. 33.
    James G., Kerber A.: The Representation Theory of the Symmetric Group. Addison-Wesley, Reading (1981)zbMATHGoogle Scholar
  34. 34.
    Jamiołkowski A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3(4), 275–278 (1972)zbMATHCrossRefADSGoogle Scholar
  35. 35.
    Kerenidis, I., Laplante, S., Lerays, V., Roland, J., Xiao, D.: Lower bounds on information complexity via zero-communication protocols and applications. In: Proceedings of FOCS, pp. 500–509. IEEE (2012). arXiv:1204.1505
  36. 36.
    Kim, I.: Conditional independence in quantum many-body systems. PhD thesis, California Institute of Technology (2013). http://thesis.library.caltech.edu/7697/
  37. 37.
    Li K., Winter A.: Relative entropy and squashed entanglement. Commun. Math. Phys. 326(1), 63–80 (2014) arXiv:1210.3181 zbMATHMathSciNetCrossRefADSGoogle Scholar
  38. 38.
    Li, K., Winter, A.: Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps (2014). arXiv:1410.4184
  39. 39.
    Lieb E.H., Ruskai M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14(12), 1938–1941 (1973)MathSciNetCrossRefADSGoogle Scholar
  40. 40.
    Müller-Lennert M., Dupuis F., Szehr O., Fehr S., Tomamichel M.: On quantum Rényi entropies: a new generalization and some properties. J. Math. Phys. 54(12), 122203 (2013) arXiv:1306.3142 MathSciNetCrossRefADSGoogle Scholar
  41. 41.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge Series on Information and the Natural Sciences. Cambridge University Press, London (2000)Google Scholar
  42. 42.
    Ogawa T., Nagaoka H.: Strong converse and Stein’s lemma in quantum hypothesis testing. IEEE Trans. Inf. Theory 46(7), 2428–2433 (2000) arXiv:quant-ph/9906090 zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Petz D.: Sufficiency of channels over von Neumann algebras. Q. J. Math. 39(1), 97–108 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Renner, R.: Security of quantum key distribution. PhD thesis, ETH Zurich (2005). arXiv:quant-ph/0512258
  45. 45.
    Renner R.: Symmetry of large physical systems implies independence of subsystems. Nat. Phys. 3, 645–649 (2007) arXiv:quant-ph/0703069 CrossRefGoogle Scholar
  46. 46.
    Renner R.: Simplifying information-theoretic arguments by post-selection. Inf. Commun. Secur. 26, 66–75 (2010)Google Scholar
  47. 47.
    Renner, R., Maurer, U.: About the mutual (conditional) information. In: Proceedings of IEEE ISIT (2002)Google Scholar
  48. 48.
    Seshadreesan, K.P., Wilde, M.M.: Fidelity of recovery and geometric squashed entanglement (2014). arXiv:1410.1441
  49. 49.
    Tomamichel, M.: A framework for non-asymptotic quantum information theory. PhD thesis, ETH Zurich (2012). arXiv:1203.2142
  50. 50.
    Tomamichel M., Colbeck R., Renner R.: A fully quantum asymptotic equipartition property. IEEE Trans. Inform. Theory 55, 5840–5847 (2009) arXiv:0811.1221 MathSciNetCrossRefGoogle Scholar
  51. 51.
    Touchette, D.: Quantum information complexity and amortized communication (2014). arXiv:1404.3733
  52. 52.
    Uhlmann A.: The “transition probability” in the state space of a *-algebra. Rep. Math. Phys. 9(2), 273–279 (1976)zbMATHMathSciNetCrossRefADSGoogle Scholar
  53. 53.
    Wang L., Renner R.: One-shot classical-quantum capacity and hypothesis testing. Phys. Rev. Lett. 108, 200501 (2012) arXiv:1007.5456 CrossRefADSGoogle Scholar
  54. 54.
    Wilde, M.M.: Quantum Information Theory. Cambridge University Press, London (2014). arXiv:1106.1445
  55. 55.
    Wilde M.M., Winter A., Yang D.: Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy. Commun. Math. Phys. 331(2), 593–622 (2014) arXiv:1306.1586 zbMATHMathSciNetCrossRefADSGoogle Scholar
  56. 56.
    Zhang L.: Conditional mutual information and commutator. Int. J. Theor. Phys. 52(6), 2112–2117 (2013) arXiv:1212.5023 zbMATHCrossRefGoogle Scholar
  57. 57.
    Zhang L., Wu J.: A lower bound of quantum conditional mutual information. J. Phys. A 47, 415303 (2014) arXiv:1403.1424 MathSciNetCrossRefGoogle Scholar

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© The Author(s) 2015

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

  1. 1.Institute for Theoretical Physics, ETH ZurichZurichSwitzerland
  2. 2.LIP, UMR 5668 ENS Lyon-CNRS-UCBL-INRIAUniversite de Lyon, ENS de LyonLyonFrance

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