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Beyond complete positivity

Quantum Information Processing volume 15, pages 1349–1360 (2016)Cite this article

Abstract

We provide a general and consistent formulation for linear subsystem quantum dynamical maps, developed from a minimal set of postulates, primary among which is a relaxation of the usual, restrictive assumption of uncorrelated initial system-bath states. We describe the space of possibilities admitted by this formulation, namely that, far from being limited to only completely positive (CP) maps, essentially any \({\mathbb {C}}\)-linear, Hermiticity-preserving, trace-preserving map can arise as a legitimate subsystem dynamical map from a joint unitary evolution of a system coupled to a bath. The price paid for this added generality is a trade-off between the set of admissible initial states and the allowed set of joint system-bath unitary evolutions. As an application, we present a simple example of a non-CP map constructed as a subsystem dynamical map that violates some fundamental inequalities in quantum information theory, such as the quantum data processing inequality.

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References

  1. Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  2. Stinespring, W.F.: Positive functions on \({C}^{*}\)-algebras. Proc. Am. Math. Soc. 6(2), 211–216 (1955)

    MathSciNet  MATH  Google Scholar 

  3. Kraus, K.: States, Effects, and Operations. Springer, Berlin (1983)

    MATH  Google Scholar 

  4. As a reminder, if \({\cal H}\) and \({\cal K}\) are Hilbert spaces, \({\cal R}\subset {\cal B}({\cal H})\) is a \({\mathbb{C}}\)-linear subspace spanned by states, and \(F:{\cal R}\rightarrow {\cal B}({\cal K})\) is \({\mathbb{C}}\)-linear, then \(F\) is completely positive if \(F\otimes \text{ id } :{\cal R}\otimes {\cal B}({\cal H}_{\text{ W }}) \rightarrow {\cal B}({\cal K})\otimes {\cal B}({\cal H}_{\text{ W }})\) is a positive map for all finite dimensional \({\cal H}_{\text{ W }}\)

  5. Rodríguez-Rosario, C.A., Modi, K., Kuah, A., Shaji, A., Sudarshan, E.C.G.: Completely positive maps and classical correlations. J. Phys. A 41(20), 205301 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Shabani, A., Lidar, D.A.: Vanishing quantum discord is necessary and sufficient for completely positive maps. Phys. Rev. Lett. 102(10), 100402 (2009)

    Article  ADS  Google Scholar 

  7. Rebentrost, P., Chakraborty, R., Aspuru-Guzik, A.: Non-Markovian quantum jumps in excitonic energy transfer. J. Chem. Phys. 131(18), 184102 (2009)

    Article  ADS  Google Scholar 

  8. Witzel, W.M., Das Sarma, S.: Quantum theory for electron spin decoherence induced by nuclear spin dynamics in semiconductor quantum computer architectures: spectral diffusion of localized electron spins in the nuclear solid-state environment. Phys. Rev. B 74, 035322 (2006)

    Article  ADS  Google Scholar 

  9. Liu, B.-H., Li, L., Huang, Y.-F., Li, C.-F., Guo, G.-C., Laine, E.-M., Breuer, H.-P., Piilo, J.: Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems. Nat. Phys. 7(12), 931–934 (2011)

    Article  Google Scholar 

  10. Štelmachovič, P., Bužek, V.: Dynamics of open quantum systems initially entangled with environment: beyond the Kraus representation. Phys. Rev. A 64, 062106 (2001)

    Article  ADS  Google Scholar 

  11. Salgado, D., Sanchez-Gomez, J.L.: Comment on “dynamics of open quantum systems initially entangled with environment: beyond the kraus representation” [pra 64, 062106 (2001)]. arXiv:quant-ph/0211164 (2002)

  12. Hayashi, H., Kimura, G., Ota, Y.: Kraus representation in the presence of initial correlations. Phys. Rev. A 67, 062109 (2003)

    Article  ADS  Google Scholar 

  13. Jordan, T.F., Shaji, A., Sudarshan, E.C.G.: Dynamics of initially entangled open quantum systems. Phys. Rev. A 70, 052110 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Salgado, D., Sánchez-Gómez, J.L., Ferrero, M.: Evolution of any finite open quantum system always admits a Kraus-type representation, although it is not always completely positive. Phys. Rev. A 70(5), 054102 (2004)

    Article  ADS  Google Scholar 

  15. Shaji, A., Sudarshan, E.C.G.: Who’s afraid of not completely positive maps? Phys. Lett. A 341(1–4), 48–54 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Carteret, H.A., Terno, D.R., Życzkowski, K.: Dynamics beyond completely positive maps: some properties and applications. Phys. Rev. A 77, 042113 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  17. Shabani, A., Lidar, D.A.: Maps for general open quantum systems and a theory of linear quantum error correction. Phys. Rev. A 80(1), 012309 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  18. Rodríguez-Rosario, C.A., Modi, K., Aspuru-Guzik, A.: Linear assignment maps for correlated system-environment states. Phys. Rev. A 81(1), 012313 (2010)

    Article  ADS  Google Scholar 

  19. Devi, A.R.U., Rajagopal, A.K., Sudha, : Open-system quantum dynamics with correlated initial states, not completely positive maps, and non-Markovianity. Phys. Rev. A 83, 022109 (2011)

    Article  ADS  Google Scholar 

  20. Modi, K., Rodríguez-Rosario, C.A., Aspuru-Guzik, A.: Positivity in the presence of initial system-environment correlation. Phys. Rev. A 86, 064102 (2012)

    Article  ADS  Google Scholar 

  21. Brodutch, A., Datta, A., Modi, K., Rivas, Á., Rodríguez-Rosario, C.A.: Vanishing quantum discord is not necessary for completely positive maps. Phys. Rev. A 87, 042301 (2013)

    Article  ADS  Google Scholar 

  22. McCracken, J.M.: Hamiltonian composite dynamics can almost always lead to negative reduced dynamics. Phys. Rev. A 88, 022103 (2013)

    Article  ADS  Google Scholar 

  23. McCracken, J.M.: Quantum channel negativity as a measure of system-bath coupling and correlation. Phys. Rev. A 88, 032103 (2013)

    Article  ADS  Google Scholar 

  24. Buscemi, F.: Complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations. Phys. Rev. Lett. 113, 140502 (2014)

    Article  ADS  Google Scholar 

  25. Tong, D.M.: Completely positive maps within the framework of direct-sum decomposition of state space. Phys. Rev. A 90, 012305 (2014)

    Article  ADS  Google Scholar 

  26. Dominy, J.M., Shabani, A., Lidar, D.A.: A general framework for complete positivity. Quantum Inf. Process. 15(1), 465–494 (2016)

  27. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, England (2000)

    MATH  Google Scholar 

  28. Mark Wilde, M.: Quantum Information Theory. Cambridge University Press, Cambride, UK (2013)

    Book  MATH  Google Scholar 

  29. Kraus, K.: General state changes in quantum theory. Ann. Phys. 64(2), 311–335 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of n-level systems. J. Math. Phys. 17(5), 821–825 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  31. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Alicki, R., Lendi, K.: Quantum Dynamical Semigroups and Applications. Lecture Notes in Physics, vol. 286. Springer, Berlin (1987)

  33. Breuer, H.-P., Laine, E.-M., Piilo, J.: Measure for the degree of non-Markovian behavior of quantum processes in open systems. Phys. Rev. Lett. 103, 210401 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  34. Laine, E.-M., Piilo, J., Breuer, H.-P.: Measure for the non-Markovianity of quantum processes. Phys. Rev. A 81, 062115 (2010)

    Article  ADS  Google Scholar 

  35. Breuer, H.-P.: Foundations and measures of quantum non-Markovianity. J. Phys. B: At. Mol. Opt. Phys. 45(15), 154001 (2012)

    Article  ADS  Google Scholar 

  36. Rodríguez-Rosario, C.A., Modi, K., Mazzola, L., Aspuru-Guzik, A.: Unification of witnessing initial system-environment correlations and witnessing non-Markovianity. EPL (Europhys. Lett.) 99(2), 20010 (2012)

    Article  ADS  Google Scholar 

  37. Bylicka, B., Chruściński, D., Maniscalco, S.: Non-markovianity and reservoir memory of quantum channels: a quantum information theory perspective. Sci. Rep. 4, 5720 (2014)

    Article  Google Scholar 

  38. Wolf, M.M., Eisert, J., Cubitt, T.S., Cirac, J.I.: Assessing non-Markovian quantum dynamics. Phys. Rev. Lett. 101, 150402 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Preskill, J.: Quantum Computation Lecture Notes Ch. 3: Foundations of Quantum Theory II: Measurement and Evolution. http://www.theory.caltech.edu/people/preskill/ph229/notes/chap3.pdf (1998)

  40. Pechukas, P.: Reduced dynamics need not be completely positive. Phys. Rev. Lett. 73(8), 1060–1062 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Nonlinear maps might also arise if one considers a nonconvex set of admissible initial system-bath states. This is developed further in [26], but a detailed analysis of the nonlinear case remains to be done

  42. Alicki, R.: Comment on reduced dynamics need not be completely positive. Phys. Rev. Lett. 75(16), 3020–3020 (1995)

    Article  ADS  Google Scholar 

  43. Pechukas, P.: Pechukas replies. Phys. Rev. Lett. 75(16), 3021–3021 (1995)

    Article  ADS  Google Scholar 

  44. Fonseca Romero, K.M., Talkner, P., Hänggi, P.: Is the dynamics of open quantum systems always linear? Phys. Rev. A 69, 052109 (2004)

    Article  ADS  Google Scholar 

  45. Note that this formulation includes no consideration as to how or why the initial state came to be in \({\cal D}_{\text{ SB }}\cap {\cal V}\); it only models how the subsystem state changes from this point forward. If one wishes to also model state preparation or other prior evolution, the map(s) representing those prior steps should be precomposed with \(\Psi _{U}^{{\cal V}}\) (and will also strongly influence the choice of \({\cal V}\)) [42]

  46. Takai, H., Yamada, H.: A note on the dilation theorems. Proc. Jpn. Acad. 48(4), 216–220 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  47. Schäffer, J.J.: On unitary dilations of contractions. Proc. Am. Math. Soc. 6(2), 322 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  48. Breuer, H.-P., Kappler, B., Petruccione, F.: Stochastic wave-function method for non-Markovian quantum master equations. Phys. Rev. A 59, 1633–1643 (1999)

    Article  ADS  Google Scholar 

  49. Breuer, H.-P.: Genuine quantum trajectories for non-Markovian processes. Phys. Rev. A 70, 012106 (2004)

    Article  ADS  Google Scholar 

  50. Budini, A.A.: Embedding non-Markovian quantum collisional models into bipartite Markovian dynamics. Phys. Rev. A 88, 032115 (2013)

    Article  ADS  Google Scholar 

  51. Hush, M.R., Lesanovsky, I., Garrahan, J.P.: Generic map from non-Lindblad to Lindblad master equations. Phys. Rev. A 91, 032113 (2015)

    Article  ADS  Google Scholar 

  52. Dajka, J., Łuczka, J.: Distance growth of quantum states due to initial system-environment correlations. Phys. Rev. A 82, 012341 (2010)

    Article  ADS  Google Scholar 

  53. Dajka, J., Łuczka, J.: The trace distance and linear entropy of qubit states: the role of initial qubit-environment correlations. Rep. Math. Phys. 70(2), 193–204 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Lindblad, G.: Completely positive maps and entropy inequalities. Commun. Math. Phys. 40, 147–151 (1975)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Uhlmann, A.: Relative entropy and the Wigner–Yanase–Dyson–Lieb concavity in an interpolation theory. Commun. Math. Phys. 54(1), 21–32 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14(12), 1938–1941 (1973)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgments

This research was supported by the ARO MURI Grant W911NF-11-1-0268. The authors thank Iman Marvian for many helpful discussions.

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Authors and Affiliations

  1. Department of Chemistry, University of Southern California, Los Angeles, CA, 90089, USA

    Jason M. Dominy & Daniel A. Lidar

  2. Department of Physics, University of Southern California, Los Angeles, CA, 90089, USA

    Daniel A. Lidar

  3. Department of Electrical Engineering, University of Southern California, Los Angeles, CA, 90089, USA

    Daniel A. Lidar

  4. Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, CA, 90089, USA

    Jason M. Dominy & Daniel A. Lidar

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  1. Jason M. Dominy
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Correspondence to Daniel A. Lidar.

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Dominy, J.M., Lidar, D.A. Beyond complete positivity. Quantum Inf Process 15, 1349–1360 (2016). https://doi.org/10.1007/s11128-015-1228-1

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