Advertisement

Quantum Information Processing

, Volume 15, Issue 1, pp 465–494 | Cite as

A general framework for complete positivity

  • Jason M. Dominy
  • Alireza Shabani
  • Daniel A. LidarEmail author
Article

Abstract

Complete positivity of quantum dynamics is often viewed as a litmus test for physicality; yet, it is well known that correlated initial states need not give rise to completely positive evolutions. This observation spurred numerous investigations over the past two decades attempting to identify necessary and sufficient conditions for complete positivity. Here, we describe a complete and consistent mathematical framework for the discussion and analysis of complete positivity for correlated initial states of open quantum systems. This formalism is built upon a few simple axioms and is sufficiently general to contain all prior methodologies going back to Pechakas (Phys Rev Lett 73:1060–1062, 1994). The key observation is that initial system-bath states with the same reduced state on the system must evolve under all admissible unitary operators to system-bath states with the same reduced state on the system, in order to ensure that the induced dynamical maps on the system are well defined. Once this consistency condition is imposed, related concepts such as the assignment map and the dynamical maps are uniquely defined. In general, the dynamical maps may not be applied to arbitrary system states, but only to those in an appropriately defined physical domain. We show that the constrained nature of the problem gives rise to not one but three inequivalent types of complete positivity. Using this framework, we elucidate the limitations of recent attempts to provide conditions for complete positivity using quantum discord and the quantum data processing inequality. In particular, we correct the claim made by two of us (Shabani and Lidar in Phys Rev Lett 102:100402–100404, 2009) that vanishing discord is necessary for complete positivity, and explain that it is valid only for a particular class of initial states. The problem remains open, and may require fresh perspectives and new mathematical tools. The formalism presented herein may be one step in that direction.

Keywords

Open quantum systems Reduced dynamics Complete positivity 

Notes

Acknowledgments

This research was supported by the ARO MURI Grant W911NF-11-1-0268 and by NSF Grant Numbers PHY-969969 and PHY-803304.

References

  1. 1.
    Kraus, K.: States, Effects, and Operations. Springer, Berlin (1983)zbMATHGoogle Scholar
  2. 2.
    Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)zbMATHGoogle Scholar
  3. 3.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  4. 4.
    Pechukas, P.: Reduced dynamics need not be completely positive. Phys. Rev. Lett. 73, 1060–1062 (1994)zbMATHMathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Pechukas, P.: A reply to the comment by Robert Alicki. Phys. Rev. Lett. 75, 3021 (1995)CrossRefADSGoogle Scholar
  6. 6.
    Jordan, T.F., Shaji, A., Sudarshan, E.C.G.: Dynamics of initially entangled open quantum systems. Phys. Rev. A 70, 052110–052114 (2004)MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    Carteret, H.A., Terno, D.R., Zyczkowski, K.: Dynamics beyond completely positive maps: some properties and applications. Phys. Rev. A 77, 042113–042118 (2008)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901–017904 (2002)CrossRefADSGoogle Scholar
  9. 9.
    Rodríguez-Rosario, C.A., Modi, K., Kuah, A., Shaji, A., Sudarshan, E.: Completely positive maps and classical correlations. J. Phys. A 41, 205301–205308 (2008)MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Shabani, A., Lidar, D.A.: Vanishing quantum discord is necessary and sufficient for completely positive maps. Phys. Rev. Lett. 102, 100402–100404 (2009)CrossRefADSGoogle Scholar
  11. 11.
    Shabani, A., Lidar, D.A.: Maps for general open quantum systems and a theory of linear quantum error correction. Phys. Rev. A 80, 012309–0123011 (2009)MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Brodutch, A., Datta, A., Modi, K., Rivas, A., Rodríguez-Rosario, C.A.: Vanishing quantum discord is not necessary for completely positive maps. Phys. Rev. A 87, 042301–042305 (2013)CrossRefADSGoogle Scholar
  13. 13.
    Buscemi, F.: Complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations. Phys. Rev. Lett. 113, 140502–140505 (2014)CrossRefADSGoogle Scholar
  14. 14.
    Alicki, R.: Comment on reduced dynamics need not be completely positive. Phys. Rev. Lett. 75, 3020 (1995)CrossRefADSGoogle Scholar
  15. 15.
    Rodríguez, C.A.: The Theory of Non-Markovian Open Quantum Systems, Ph.D. thesis, The University of Texas at Austin (2008)Google Scholar
  16. 16.
    Modi, K., Rodríguez-Rosario, C.A., Aspuru-Guzik, A.: Positivity in the presence of initial system-environment correlation. Phys. Rev. A 86, 064102–064105 (2012)CrossRefADSGoogle Scholar
  17. 17.
    Rodríguez-Rosario, C.A., Modi, K., Aspuru-Guzik, A.: Linear assignment maps for correlated system-environment states. Phys. Rev. A 81, 012313–012315 (2010)CrossRefADSGoogle Scholar
  18. 18.
    Štelmachovič, P., Bužek, V.: Dynamics of open quantum systems initially entangled with environment: beyond the Kraus representation. Phys. Rev. A 64, 062106-5 (2001)ADSGoogle Scholar
  19. 19.
    Romero, K.M.F., Talkner, P., Hänggi, P.: Is the dynamics of open quantum systems always linear? Phys. Rev. A 69, 052109-8 (2004)CrossRefADSGoogle Scholar
  20. 20.
    Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants. Springer, New York (2009)zbMATHCrossRefGoogle Scholar
  21. 21.
    Hayashi, H., Kimura, G., Ota, Y.: Kraus representation in the presence of initial correlations. Phys. Rev. A 67, 062109-5 (2003)ADSGoogle Scholar
  22. 22.
    Grace, M.D., Dominy, J., Kosut, R.L., Brif, C., Rabitz, H.: Environment-invariant measure of distance between evolutions of an open quantum system. New J. Phys. 12, 015001–0150012 (2010). special Issue: Focus on Quantum ControlMathSciNetCrossRefADSGoogle Scholar
  23. 23.
    Salgado, D., Sanchez-Gomez, J.L.: arXiv:quant-ph/0211164
  24. 24.
    de Pillis, J.: Linear transformations which preserve hermitian and positive semidefinite operators. Pac. J. Math. 23, 129–137 (1967)zbMATHCrossRefADSGoogle Scholar
  25. 25.
    Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebr. Appl. 10, 285–290 (1975)zbMATHCrossRefGoogle Scholar
  26. 26.
    Stinespring, W.F.: Positive functions on C*-algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Jenčová, A.: Generalized channels: channels for convex subsets of the state space. J. Math. Phys. 53, 012201–0122023 (2012)MathSciNetCrossRefADSGoogle Scholar
  28. 28.
    Arveson, W.: Subalgebras ofC \(C^*\)-algebras. Acta Math. 123, 141–141 (1969)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Heinosaari, T., Jivulescu, M.A., Reeb, D., Wolf, M.M.: Extending quantum operations. J. Math. Phys. 53, 102208–1022029 (2012)MathSciNetCrossRefADSGoogle Scholar
  30. 30.
    Kraus, K.: General state changes in quantum theory. Ann. Phys. 64, 311–335 (1971)zbMATHMathSciNetCrossRefADSGoogle Scholar
  31. 31.
    Choi, M.-D., Effros, E.G.: Injectivity and operator spaces. J. Funct. Anal. 24, 156–209 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Araki, H., Lieb, E.H.: Entropy inequalities. Comm. Math. Phys. 18, 160–170 (1970)MathSciNetCrossRefADSGoogle Scholar
  33. 33.
    Bruch, L.W., Falk, H.: Gibbs inequality in quantum statistical mechanics. Phys. Rev. A 2, 1598–1599 (1970)MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jason M. Dominy
    • 1
    • 4
  • Alireza Shabani
    • 5
    • 6
  • Daniel A. Lidar
    • 1
    • 2
    • 3
    • 4
    Email author
  1. 1.Department of ChemistryUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of Electrical EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  3. 3.Department of PhysicsUniversity of Southern CaliforniaLos AngelesUSA
  4. 4.Center for Quantum Information Science and TechnologyUniversity of Southern CaliforniaLos AngelesUSA
  5. 5.Department of ChemistryUniversity of CaliforniaBerkeleyUSA
  6. 6.Google ResearchVeniceUSA

Personalised recommendations