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Review: Characterizing and quantifying quantum chaos with quantum tomography

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Abstract

We explore quantum signatures of classical chaos by studying the rate of information gain in quantum tomography. The tomographic record consists of a time series of expectation values of a Hermitian operator evolving under the application of the Floquet operator of a quantum map that possesses (or lacks) time-reversal symmetry. We find that the rate of information gain, and hence the fidelity of quantum state reconstruction, depends on the symmetry class of the quantum map involved. Moreover, we find an increase in information gain and hence higher reconstruction fidelities when the Floquet maps employed increase in chaoticity. We make predictions for the information gain and show that these results are well described by random matrix theory in the fully chaotic regime. We derive analytical expressions for bounds on information gain using random matrix theory for different classes of maps and show that these bounds are realized by fully chaotic quantum systems.

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References

  1. Steven H Strogatz, Nonlinear dynamics and chaos (Westview Press, 1994)

  2. I Ispalatov, V Madhok, S Allende and M Doebeli, Sci. Rep. 5, Article number: 12506

  3. B O Koopman, Proceedings of the National Academy of Sciences 17(5), 315 (1931)

  4. F Haake, Quantum signatures of chaos (Springer-Verlag, Berlin, 1991)

    Book  MATH  Google Scholar 

  5. F Dyson, J. Math. Phys. 3, 140 (1962)

    Article  ADS  MathSciNet  Google Scholar 

  6. O Bohigas, M J Giannoni, and C Schmit, Phys. Rev. Lett. 52, 1 (1984)

  7. A Peres, Phys. Rev. A 30, 1610 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  8. R Schack and C Caves, Phys. Rev. E 53, 3257 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  9. K M Frahm, R Fleckinger, and D L Shepelyansky, Eur. Phys. J. D 29, 139 (2004)

    Article  ADS  Google Scholar 

  10. T Prosen and M Znidaric, Phys. Rev. E 75, 015202 (2007)

    Article  ADS  Google Scholar 

  11. W Zurek and J Paz, Phys. Rev. Lett. 72, 2508 (1994)

    Article  ADS  Google Scholar 

  12. J Emerson, Y S Weinstein, M Saraceno, S Lloyd, and D G Cory, Science 302, 2098 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  13. K Furuya, M C Nemes, and G Q Pellegrino, Phys. Rev. Lett. 80, 5524 (1998)

    Article  ADS  Google Scholar 

  14. A Lakshminarayan, Phys. Rev. E 64, 036207 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  15. J Bandyopadhyay and A Lakshminarayan, Phys. Rev. Lett. 89, 60402 (2002)

    Article  ADS  Google Scholar 

  16. A Lakshminarayan, Phys. Rev. E 64, 36207 (2001)

    Article  ADS  Google Scholar 

  17. P Miller and S Sarkar, Phys. Rev. E 60, 1542 (1999)

    Article  ADS  Google Scholar 

  18. S Ghose and B C Sanders, Phys. Rev. A 70, 062315 (2004)

    Article  ADS  Google Scholar 

  19. X Wang, S Ghose, B Sanders, and B Hu, Phys. Rev. E 70, 16217 (2004)

    Article  ADS  Google Scholar 

  20. J Bandyopadhyay and A Lakshminarayan, Phys. Rev. R 69, 16201 (2004)

    ADS  MathSciNet  Google Scholar 

  21. Ph Jacquod, Phys. Rev. Lett. 92, 150403 (2004) C Petitjean and Ph Jacquod, Phys. Rev. Lett. 97, 194103 (2006)

  22. C M Trail, V Madhok, and I H Deutsch, Phys. Rev. E 76, 046211 (2008)

    Article  ADS  Google Scholar 

  23. V Madhok, V Gupta, D Trottier, and S Ghose, Phys. Rev. E 91, 032906 (2015)

    Article  ADS  Google Scholar 

  24. V Madhok, Phys. Rev. E 92, 036901 (2015)

    Article  ADS  Google Scholar 

  25. E Lubkin, J. Math. Phys. 19, 1028 (1978)

    Article  ADS  Google Scholar 

  26. D N Page, Phys. Rev. Lett. 71, 1291 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  27. S Sen, Phys. Rev. Lett. 77, 1 (1996)

    Article  ADS  Google Scholar 

  28. K Zyczkowski and H -J Sommers, J. Phys. A: Math. Gen. 34, 7111 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  29. V Cappellini, H J Sommers, and K Zyczkowski, Phys. Rev. A 74, 062322 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  30. H Kubotani, S Adachi, and M Toda, Phys. Rev. Lett. 100, 240501 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  31. S Kumar and A Pandey, J. Phys. A 44, 445301 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  32. V Madhok, C Riofrío, S Ghose, and I H Deutsch, Phys. Rev. Lett. 112, 014102 (2014)

    Article  ADS  Google Scholar 

  33. A Silberfarb, P Jessen, and I H Deutsch, Phys. Rev. Lett. 95, 030402 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  34. S Chaudhury, A Smith, B E Anderson, S Ghose, and P Jessen, Nature 461, 768 (2009)

    Article  ADS  Google Scholar 

  35. A Smith, C A Riofrío, B E Anderson, H Sosa-Martinez, I H Deutsch, and P S Jessen, Phys. Rev. A 87, 030102(R) (2013)

    Article  ADS  Google Scholar 

  36. C A Riofrío, P S Jessen, and I H Deutsch, J. Phys. B 44, 154007

  37. S T Merkel, C A Riofrío, S T Flammia, and I H Deutsch, Phys. Rev. A 81, 032126 (2010)

    Article  ADS  Google Scholar 

  38. A Ben-Israel and T N E Greville, Generalized inverses: Theory and applications (Springer-Verlag, New York, 2003)

    MATH  Google Scholar 

  39. Č Brukner and A Zeilinger, Phys. Rev. Lett. 83, 3354 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  40. J Řeháček and Z Hradil, Phys. Rev. Lett. 88, 130401 (2002)

    Article  MathSciNet  Google Scholar 

  41. T Cover and J Thomas, Elements of information theory (Wiley & Sons, New York, 2006)

    MATH  Google Scholar 

  42. M Kus, J Mostowski, and F Haake, J. Phys. A 21, L1073

  43. C H Baldwin, I H Deutsch, and A Kalev, Phys. Rev. A 93, 052105 (2016)

    Article  ADS  Google Scholar 

  44. A Kalev, R L Kosut and I H Deutsch, arXiv:1508.00536(2012)

  45. W Wootters, Found. Phys. 20, 1365 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  46. Ya G Sinai, Dokl. Russ. Acad. Sci. 124, 768 (1959)

    Google Scholar 

  47. R L Cook, C A Riofrío, and I H Deutsch, Phys. Rev. A 90, 032113

  48. A Scott and C Caves, J. Phys. A 36, 9553 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  49. A Peres, Phys. Rev. A 30, 1610 (1984)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

The authors acknowledge useful discussions with Prof. Shohini Ghose and Prof. Arul Lakshminarayan. VM acknowledges support from NSERC, Canada (through Discovery grant to M Doebeli). CR acknowledges the support by the Freie Universität Berlin within the Excellence Initiative of the German Research Foundation. IHD acknowledges support of the US National Science Foundation, Grants, PHY-1212445 and PHY-1307520.

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Authors and Affiliations

  1. Department of Zoology, University of British Columbia, 6270 University Boulevard, Vancouver, V6T 1Z4, Canada

    VAIBHAV MADHOK

  2. Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195, Berlin, Germany

    CARLOS A RIOFRÍO

  3. Department of Physics and Astronomy, Room 24, University of New Mexico, 800 Yale Blvd., Albuquerque, NM, 87131-1156, Mexico

    IVAN H DEUTSCH

Authors
  1. VAIBHAV MADHOK
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  2. CARLOS A RIOFRÍO
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  3. IVAN H DEUTSCH
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Correspondence to VAIBHAV MADHOK.

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MADHOK, V., RIOFRÍO, C.A. & DEUTSCH, I.H. Review: Characterizing and quantifying quantum chaos with quantum tomography. Pramana - J Phys 87, 65 (2016). https://doi.org/10.1007/s12043-016-1259-x

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  • DOI: https://doi.org/10.1007/s12043-016-1259-x

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