Universal Preparability of States and Asymptotic Completeness

Abstract

We introduce a notion of universal preparability for a state of a system, more precisely: for a normal state on a von Neumann algebra. It describes a situation where from an arbitrary initial state it is possible to prepare a target state with arbitrary precision by a repeated interaction with a sequence of copies of another system. For \({\mathcal{B}(\mathcal{H})}\) we give criteria sufficient to ensure that all normal states are universally preparable which can be verified for a class of non-commutative birth and death processes realized, in particular, by the interaction of a micromaser with a stream of atoms. As a tool, the theory of tight sequences of states and of stationary states is further developed and we show that in the presence of stationary faithful normal states universal preparability of all normal states is equivalent to asymptotic completeness, a notion studied earlier in connection with the scattering theory of non-commutative Markov processes.

References

  1. Ba75

    Barnett, S.: Introduction to Mathematical Control Theory. Oxford University Press, (1975)

  2. BGKRSS

    Bücher D., Gärtner A., Kümmerer B., Reußwig W., Schwieger K., Sissouno N.: Ergodic properties of quantum birth and death chains. Preprint, arXiv:1306.3776 (2013)

  3. BG07

    Burgarth D., Giovannetti V.: Full control by locally induced relaxation. Phys. Rev. Lett. 99, 100501 (2007)

    ADS  Article  Google Scholar 

  4. Co07

    Coron J.M., Control and Nonlinearity. AMS Mathematical Surveys and Monographs, vol. 136 (2007)

  5. Da76

    Davies E.B.: Quantum Theory of Open Systems. Academic Press, London (1976)

    Google Scholar 

  6. DMR08

    Davidson K.R., Marcoux L.W., Radjavi H.: Transitive spaces of operators. Integral Equ. Oper. Theory 61(2), 187–210 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  7. FR01

    Fagnola F., Rebolledo R.: On the existence of stationary states for quantum dynamical semigroups. J. Math. Phys. 42(3), 1296–1308 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. FR02

    Fagnola F., Rebolledo R.: Subharmonic projections for a quantum Markov semigroup. J. Math. Phys. 43(2), 1074–1082 (2002)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. GK12

    Gärtner A., Kümmerer B.: A coherent approach to recurrence and transience for quantum Markov operators. Preprint, arXiv:1211.6876 (2012)

  10. Go04

    Gohm R.: Noncommutative Stationary Processes, vol. 1839. Springer LNM, Berlin (2004)

  11. Go04b

    Gohm R.: Kümmerer-Maassen scattering theory and entanglement. In: Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 7(2), pp. 271–280. World Scientific, Singapore (2004)

  12. Go09

    Gohm R.: Noncommutative Markov chains and multi-analytic operators. J. Math. Anal. Appl. 364(1), 275–288 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  13. Go15

    Gohm, R.: Weak Markov processes as linear systems. In: Mathematics of Control, Signals, and Systems (MCSS), pp. 375–413 (2015)

  14. GKL06

    Gohm R., Kümmerer B., Lang T.: Noncommutative symbolic coding. Ergod. Theory Dyn. Syst. 26, 1521–1548 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  15. GK82

    Groh U., Kümmerer B.: Bibounded operators on W*-algebras. Math. Scand. 50, 269–285 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  16. Ha06

    Haag, F.: Asymptotisches Verhalten von Quanten–Markov–Halbgruppen und Quanten–Markov–Prozessen, Dissertation, Darmstadt (2006)

  17. HR06

    Haroche S., Raimond J.M.: Exploring the Quantum. Atoms, Cavities and Photons. Oxford University Press, Oxford (2006)

    Google Scholar 

  18. Kr85

    Krengel, U.: Ergodic Theorems. De Gruyter, Berlin (1985)

  19. Ku85

    Kümmerer B.: Markov dilations on W*-algebras. J. Funct. Anal. 63, 139–177 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  20. Ku06

    Kümmerer B.: Quantum Markov processes and applications in physics. In: Barndorff-Nielsen, O., Franz, U., Gohm, R., Kümmerer, B., Thorbjørnsen, S. (eds.) Quantum Independent Increment Processes II, Springer Lecture Notes in Mathematics, vol. 1866, pp. 259–330 (2006)

  21. Ku08

    Kümmerer, B.: Asymptotic behaviour of quantum Markov processes. in: Infinite Dimensional Harmonic Analysis IV, pp. 168–183 (2008)

  22. KM00

    Kümmerer B., Maassen H.: A scattering theory for Markov chains. Inf. Dimens. Anal. Quantum Prob. Rel. Topics 3, 161–176 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  23. KN79

    Kümmerer B., Nagel R.: Mean ergodic semigroups on \({W^*}\)-algebras. Acta Sci. Math. 41, 151–159 (1979)

    MathSciNet  MATH  Google Scholar 

  24. KR86

    Kadison R.V., Ringrose J.R.: Fundamentals of the Theory of Operator Algebras. Academic Press, London (1986)

    Google Scholar 

  25. Lu95

    Luczak A.: Ergodic projection for quantum dynamical semigroups. Int. J. Theor. Phys. 34(8), 1533–1540 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  26. MS91

    Meystre P., Sargent M.: Elements of Quantum Optics. Springer, Berlin (1991)

    Google Scholar 

  27. Ro14

    Rouchon P.: Models and Feedback Stabilization of Open Quantum Systems. Extended version of paper for ICM Seoul (v3). arXiv:1407.7810 (2014)

  28. S-H11

    Sayrin C., Dotsenko I., Zhou X., Peaudecerf B., Rybarczyk T., Gleyzes S., Rouchon P., Mirrahimi M., Amini H., Brune M., Raimond J.M., Haroche S.: Real-time quantum feedback prepares and stabilizes photon number states. Nature 477, 73–77 (2011)

    ADS  Article  Google Scholar 

  29. Sh96

    Shiryaev, A.N.: Probability. Springer, Berlin (1996)

  30. St13

    Størmer E.: Positive Linear Maps of Operator Algebras. Springer, Berlin (2013)

    Google Scholar 

  31. Tak72

    Takesaki M.: Conditional expectations in von Neumann algebras. J. Funct. Anal. 9, 306–321 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  32. Tak79

    Takesaki, M.: Theory of Operator Algebras I. Springer, 2nd printing of the First Edition (1979)

  33. WBKM00

    Wellens T., Buchleitner A., Kümmerer B., Maassen H.: Quantum state preparation via asymptotic completeness. Phys. Rev. Lett. 85, 3361–3364 (2000)

    ADS  Article  Google Scholar 

  34. Wi07

    Wiskandt J.: Asymptotische Vollständigkeit und Beobachtbarkeit von Quanten–Markov–Prozessen. Diplomarbeit, Darmstadt (2007)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Rolf Gohm.

Additional information

Communicated by M. M. Wolf

Rights and permissions

Open Access This 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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gohm, R., Haag, F. & Kümmerer, B. Universal Preparability of States and Asymptotic Completeness. Commun. Math. Phys. 352, 59–94 (2017). https://doi.org/10.1007/s00220-017-2851-8

Download citation