Skip to main content
SpringerLink
Log in

Equivalence Classes and Local Asymptotic Normality in System Identification for Quantum Markov Chains

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the problem of identifying and estimating dynamical parameters of an ergodic quantum Markov chain, when only the stationary output is accessible for measurements. The starting point of the analysis is the fact that the knowledge of the output state completely fixes the dynamics up to an equivalence class of ‘coordinate transformation’ consisting of a multiplication by a phase and a unitary conjugation of the Kraus operators.

Assuming that the dynamics depends on an unknown parameter, we show that the latter can be estimated at the ‘standard’ rate n −1/2, and give an explicit expression of the (asymptotic) quantum Fisher information of the output, which is proportional to the Markov variance of a certain ‘generator’. More generally, we show that the output is locally asymptotically normal, i.e., it can be approximated by a simple quantum Gaussian model consisting of a coherent state whose mean is related to the unknown parameter. As a consistency check, we prove that a parameter related to the ‘coordinate transformation’ unitaries has zero quantum Fisher information.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baumgartner B., Narnhofer H.: Analysis of quantum semigroups with GKSLindblad generators: II General. J. Phys. A: Math. Theor. 41, 395303 (2008)

    Article  MathSciNet  Google Scholar 

  2. Baumgratz T., Gross D., Cramer M., Plenio M.B.: Scalable reconstruction of density matrices. Phys. Rev. Lett. 111, 020401 (2013)

    Article  ADS  Google Scholar 

  3. Belavkin V.P.: Generalized Heisenberg uncertainty relations, and efficient measurements in quantum systems. Theor. Math. Phys. 26, 213–222 (1976)

    Article  MathSciNet  Google Scholar 

  4. Bickel P.J., Ritov Y., Ryden T.: Inference in hidden Markov models I: local asymptotic normality in the stationary case. Bernoulli 2, 199–291 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bickel P.J., Ritov Y., Ryden T.: Asymptotic normality of the maximum likelihood estimator for general hidden Markov models. Ann. Stat. 26, 1614–1635 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Burgarth D., Maruyama K., Nori F.: Indirect quantum tomography of quadratic Hamiltonians. New J. Phys. 13, 013019 (2011)

    Article  ADS  Google Scholar 

  7. Catana C., Guţă M.: Heisenberg versus standard scaling in quantum metrology with Markov generated states and monitored environment. Phys. Rev. A 90, 012330 (2014)

    Article  ADS  Google Scholar 

  8. Catana, C., Guţă, M., Bouten, L.: Fisher informations and local asymptotic normality for continuous-time quantum Markov processes. arXiv:1407.5131

  9. Catana C., Guţă M., Kypraios T.: Maximum likelihood versus likelihood-free quantum system identification in the atom maser. J. Phys. A: Math. Theor. 47, 415302 (2014)

    Article  Google Scholar 

  10. Catana C., van Horssen M., Guţă M.: Asymptotic inference in system identification for the atom maser. Phil. Trans. R. Soc. Lond. A 370, 5308–5323 (2012)

    Article  ADS  Google Scholar 

  11. Chefles A., Josza R., Winter A.: On the existence of physical transformations between sets of quantum states. Int. J. Quantum Inf. 02, 11 (2004)

    Article  Google Scholar 

  12. Cole J., Schirmer S., Greentree A., Wellard C., Oi D., Hollenberg L.: Identifying an experimental two-state Hamiltonian to arbitrary accuracy. Phys. Rev. A 71, 062312 (2005)

    Article  ADS  Google Scholar 

  13. Douc R., Matias C.: Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7, 381–420 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dowling J.P., Milburn G.J.: Quantum technology: the second quantum revolution. Phil. Trans. R. Soc. Lond. A 361, 1655–1674 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  15. Evans D.E., Hoegh-Krohn R.: Spectral properties of positive maps on c *-algebras. J. Lond. Math. Soc. s2-17, 345 (1978)

    Article  MathSciNet  Google Scholar 

  16. Fannes M., Nachtergaele B., Werner R.F.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144, 443 (1992)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Fannes M., Nachtergaele B., Werner R.F.: Finitely correlated pure states. J. Funct. Anal. 120, 511–534 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fujiwara A.: Quantum channel identification problem. Phys. Rev. A 63, 042304 (2001)

    Article  ADS  Google Scholar 

  19. Gardiner C., Zoller P.: Quantum Noise. Springer, Berlin (2004)

    MATH  Google Scholar 

  20. Gill R.D., Guţă M.: On asymptotic quantum statistical inference. Inst. Math. Stat. Collect. 9, 105–127 (2012)

    Google Scholar 

  21. Guţă M., Jençová A.: Local asymptotic normality in quantum statistics. Commun. Math. Phys. 276, 341–379 (2007)

    Article  ADS  MATH  Google Scholar 

  22. Guţă M., Kahn J.: Local asymptotic normality for qubit states. Phys. Rev. A 73, 052108 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  23. Guţă, M., van Horssen, M.: Large deviations, central limit and dynamical phase transitions in the atom maser. arXiv:1206.4956v2 (2012)

  24. Guţă M.: Fisher information and asymptotic normality in system identification for quantum Markov chains. Phys. Rev. A 83, 062324 (2011)

    Article  ADS  Google Scholar 

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

    Book  Google Scholar 

  26. Helstrom C.W.: Quantum Detection and Estimation Theory. Academic Press, New York (1976)

    MATH  Google Scholar 

  27. Holevo A.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Publishing Company, Amsterdam (1982)

    MATH  Google Scholar 

  28. Holstein T., Primakoff H.: Field dependence of the intrinsic domain magnetization of a ferromagnet. Phys. Rev. 58, 1098–1113 (1940)

    Article  ADS  MATH  Google Scholar 

  29. Howard M., Twamley J., Wittmann C., Gaebel T., Jelezko F., Wrachtrup J.: Quantum process tomography and Linblad estimation of a solid-state qubit. New J. Phys. 8, 33 (2006)

    Article  ADS  Google Scholar 

  30. Kahn J., Guţă M.: Local asymptotic normality for finite dimensional quantum systems. Commun. Math. Phys. 289, 597–652 (2009)

    Article  ADS  MATH  Google Scholar 

  31. Lesanovsky I., Garrahan J.P.: Thermodynamics of quantum jump trajectories. Phys. Rev. Lett. 104, 160601 (2010)

    Article  Google Scholar 

  32. Lesanovsky I., van Horssen M., Guţă M., Garrahan J.P.: Characterization of dynamical phase transitions in quantum jump trajectories beyond the properties of the stationary state. Phys. Rev. Lett. 110, 150401 (2013)

    Article  ADS  Google Scholar 

  33. Mabuchi H., Khaneja N.: Principles and applications of control in quantum systems. Int. J. Robust Nonlinear Control 15, 647–667 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Pérez-García D., Verstraete F., Wolf M.M., Cirac J.I.: Matrix product state representations. Quantum Inf. Comput. 7, 401 (2007)

    MATH  MathSciNet  Google Scholar 

  35. Petrie T.: Probabilistic functions of finite state Markov chains. Ann. Math. Stat. 40, 97–115 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  36. Petz D., Jencova A.: Sufficiency in quantum statistical inference. Commun. Math. Phys. 263, 259–276 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Sanz M., Pérez-García D., Wolf M.M., Cirac J.I.: A Quantum version of Wielandt’s inequality. IEEE Trans. Inform. Theory 56, 4668–4673 (2010)

    Article  MathSciNet  Google Scholar 

  38. van der Vaart A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  39. Wiseman H.M., Milburn G.J.: Quantum Measurement and Control. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Madalin Guta & Jukka Kiukas

Authors
  1. Madalin Guta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jukka Kiukas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Madalin Guta.

Additional information

Communicated by M. M. Wolf

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guta, M., Kiukas, J. Equivalence Classes and Local Asymptotic Normality in System Identification for Quantum Markov Chains. Commun. Math. Phys. 335, 1397–1428 (2015). https://doi.org/10.1007/s00220-014-2253-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2253-0

Keywords

Search

Navigation