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Evaluation of Estimation Precision in Quantum Tomography

  • Takanori Sugiyama
Chapter
Part of the Springer Theses book series (Springer Theses)

Abstract

Quantum tomography is a general term for estimation methods used to completely identify quantum states or processes using independent experimental designs, and has become a standard measurement technique in quantum physics. It is especially important in the field of quantum information as it is used for the confirmation of successful experimental implementation of quantum protocols. For example, it can be used to confirm that the quantum states produced in a quantum information protocol are sufficiently close to their theoretical targets. In spite of this importance, however, finite sample analysis in quantum tomography has not been well studied. In this chapter, we explain our results regarding finite sample analysis of quantum tomography. In Sect. 5.1, we explain the estimation setting. In Sect. 5.2, we analyze expected losses with finite data, particularly for three estimators: extended linear, extended norm-minimization, and maximum-likelihood. In Sect. 5.3, we derive upper bounds on error probability with finite data for those same estimators.

Keywords

Quantum tomography Extended norm-minimization estimator Maximum-likelihood estimator 

References

  1. 1.
    D.T. Smithey, M. Beck, M.G. Raymer, A. Faridani, Phys. Rev. Lett. 70, 1244 (1993). doi: 10.1103/PhysRevLett.70.1244 Google Scholar
  2. 2.
    Z. Hradil, Phys. Rev. A 55, R1561 (1997). doi: 10.1103/PhysRevA.55.R1561
  3. 3.
    K. Banaszek, G.M. D’Ariano, M.G.A. Paris, M.F. Sacchi, Phys. Rev. A 61, 010304(R) (1999). doi: 10.1103/PhysRevA.61.010304
  4. 4.
    J.F. Poyatos, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 78, 390 (1997). doi: 10.1103/PhysRevLett.78.390 Google Scholar
  5. 5.
    I.L. Chuang, M.A. Nielsen, J. Mod. Phys. 44, 2455 (1997). doi: 10.1080/09500349708231894
  6. 6.
    V. Buzek, Phys. Rev. A 58, 1723 (1998). doi: 10.1103/PhysRevA.58.1723
  7. 7.
    J. Fiurasek, Z. Hradil, Phys. Rev. A 63, 020101(R) (2001). doi: 10.1103/PhysRevA.63.020101
  8. 8.
    M.F. Sacchi, Phys. Rev. A 63, 054104 (2001). doi: 10.1103/PhysRevA.63.054104
  9. 9.
    A. Luis, L.L. Sanchez-Sato, Phys. Rev. Lett. 83, 3573 (1999). doi: 10.1103/PhysRevLett.83.3573 Google Scholar
  10. 10.
    J. Fiurasek, Phys. Rev. A 64, 024102 (2001). doi: 10.1103/PhysRevA.64.024102
  11. 11.
    G.M. D’Ariano, P.L. Presti, Phys. Rev. Lett. 86, 4195 (2001). doi: 10.1103/PhysRevLett.86.4195 Google Scholar
  12. 12.
    F. Bloch, Phys. Rev. 70, 460 (1946). doi: 10.1103/PhysRev.70.460
  13. 13.
    E. Bagan, M. Baig, R. Muñoz-Tapia, A. Rodriguez, Phys. Rev. A 69, 010304(R) (2004). doi: 10.1103/PhysRevA.69.010304
  14. 14.
    G. Kimura, Phys. Lett. A 314, 339 (2003). doi: 10.1016/S0375-9601(03)00941–1
  15. 15.
    M.S. Byrd, N. Khaneja, Phys. Rev. A 68, 062322 (2003). doi: 10.1103/PhysRevA.68.062322
  16. 16.
    U. Fano, Rev. Mod. Phys. 29, 74 (1957). doi: 10.1103/RevModPhys.29.74
  17. 17.
    R. Schack, T.A. Brum, C.M. Caves, Phys. Rev. A 64, 014305 (2001). doi: 10.1103/PhysRevA.64.014305
  18. 18.
    C.A. Fuchs, R. Schack, P.F. Scudo, Phys. Rev. A 69, 062305 (2004). doi: 10.1103/PhysRevA.69.062305
  19. 19.
    V. Buzěk, G. Drobny, J. Mod. Opt. 47, 2823 (2000). doi: 10.1080/09500340008232199 Google Scholar
  20. 20.
    S.T. Flammia, D. Gross, Y.K. Liu, J. Eisert, New J. Phys. 14, 095022 (2012). doi: 10.1088/1367-2630/14/9/095022 Google Scholar
  21. 21.
    R. Blume-Kohout, arXiv:1202.5270 [quant-ph] (2012).Google Scholar
  22. 22.
    M. Christandl, R. Renner, Phys. Rev. Lett. 109, 120403 (2012). doi: 10.1103/PhysRevLett.109.120403 Google Scholar
  23. 23.
    A.J. Scott, J. Phys. A: Math. Gen. 39, 13507 (2006). doi: 10.1088/0305-4470/39/43/009
  24. 24.
    H. Zhu, B.G. Englert, Phys. Rev. A 84, 022327 (2011). doi: 10.1103/PhysRevA.84.022327
  25. 25.
    M.D. de Burgh, N.K. Langford, A.C. Doherty, A. Gilchrist, Phys. Rev. A 78, 052122 (2008). doi: 10.1103/PhysRevA.78.052122
  26. 26.
    T. Sugiyama, P.S. Turner, M. Murao, Phys. Rev. A 85, 052107 (2012). doi: 10.1103/PhysRevA.85.052107
  27. 27.
    H. Chernoff, Ann. Math. Stat. 25, 573 (1954). doi: 10.1214/aoms/1177728725
  28. 28.
    S.G. Self, K.Y. Liang, J. Am. Stat. Assoc. 82, 605 (1987). doi: 10.1080/01621459.1987.10478472 Google Scholar
  29. 29.
    M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Wiley, New York, 1972)Google Scholar
  30. 30.
    J.A. Smolin, J.M. Gambetta, G. Smith, Phys. Rev. Lett. 108, 070502 (2012). doi: 10.1103/PhysRevLett.108.070502 Google Scholar
  31. 31.
    S.M. Tan, J. Mod. Opt. 44, 2233 (1997). doi: 10.1080/09500349708231881
  32. 32.
    C.W. Helstrom, Quantum Detection and Estimation Theory (Academic, New Tork, 1976)Google Scholar
  33. 33.
    A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, New York, 1982)Google Scholar
  34. 34.
    M. Hayashi (ed.), Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, Singapore, 2005)Google Scholar
  35. 35.
    K. Vogel, H. Risken, Phys. Rev. A 40, 2847 (1989). doi: 10.1103/PhysRevA.40.2847
  36. 36.
    T.M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-Beam CT (Springer, Berlin, 2008)Google Scholar
  37. 37.
    M. Paris, J. Řeháček (eds.), Quantum State Estimation. Lecture Notes in Physics (Springer, Berlin, 2004)Google Scholar
  38. 38.
    A. Ling, K.P. Soh, A. L.-Linares, C. Kurtsiefer, Phys. Rev. A 74, 022309 (2006). doi: 10.1103/PhysRevA.74.022309
  39. 39.
    H. Kosaka, T. Inagaki, Y. Rikitake, H. Imamura, Y. Mitsumori, K. Edamatsu, Nature 457, 702 (2009). doi: 10.1038/nature07729
  40. 40.
    M. Steffen, M. Ansmann, R. McDermott, N. Katz, R.C. Bialczak, E. Lucero, Phys. Rev. Lett. 97, 050502 (2006). doi: 10.1103/PhysRevLett.97.050502 Google Scholar
  41. 41.
    M. Neeley, M. Ansmann, R.C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A.N. Cleland, J.M. Martinis, Nature Phys. 4, 523 (2008). doi: 10.1038/nphys972
  42. 42.
    M. Hofheinz, H. Wang, M. Ansmann, R.C. Bialczak, E. Lucero, M. Neeley, A.D. O’Connell, D. Sank, J. Wenner, J.M. Martinis, A.N. Cleland, Nature 459, 546 (2009). doi: 10.1038/nature08005
  43. 43.
    D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 77, 4281 (1996). doi: 10.1103/PhysRevLett.77.4281 Google Scholar
  44. 44.
    S. Olmschens, D.N. Matsukevich, P. maunz, D. hayes, L.M. Duan, C. Monroe, Science 323, 486 (2009). doi: 10.1126/science.1167209
  45. 45.
    H. Tanji, S. Ghosh, J. Simon, B. Bloom, V. Vuletic, Phys. Rev. Lett. 103, 043601 (2009). doi: 10.1103/PhysRevLett.103.043601 Google Scholar
  46. 46.
    P. Neumann, N. Mizouchi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, J. Wrachtrup, Science 320, 1326 (2008). doi: 10.1126/science.1157233
  47. 47.
    T.J. Dunn, I.A. Walmsley, S. Mukamel, Phys. Rev. Lett. 74, 884 (1995). doi: 10.1103/PhysRevLett.74.884 Google Scholar
  48. 48.
    M.A. Nielsen, E. Knill, R. Laflamme, Nature 396, 52 (1998). doi: 10.1038/23891
  49. 49.
    E. Skovsen, H. Stapelfeldt, S. Juhl, K. Molmer, Phys. Rev. Lett. 91, 090406 (2003). doi: 10.1103/PhysRevLett.91.090406 Google Scholar
  50. 50.
    R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, New York, 1985)Google Scholar
  51. 51.
    T.M. Cover, J.A. Thomas, Elements of Information Theory, 2nd edn. Wiley Series in Telecommunications and Signal Processing (Wiley-Interscience, New York, 2006)Google Scholar
  52. 52.
    I. Bengtsson, K. Życzkowski, Geometry of Quantum States (Cambridge University Press, Cambridge, 2006)Google Scholar
  53. 53.
    J.M. Borwein, A.S. Lewis, Convex Analysis and Nonlinear Optimization. CMS Books in Mathematics (Springer, New York, 2006)Google Scholar
  54. 54.
    N.J. Higham, Accuracy and Stability of Numerical Algorithms (SIAM., Philadelphia, 2002).Google Scholar
  55. 55.
    T. Sugiyama, P.S. Turner, M. Murao, Phys. Rev. Lett. 111, 160406 (2013). doi: 10.1103/PhysRevLett.111.160406 Google Scholar
  56. 56.
    T. Sugiyama, P.S. Turner, M. Murao, New J. Phys. 14, 085005 (2012). doi: 10.1088/1367-2630/14/8/085005 Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Department of Physics Graduate School of ScienceThe University of TokyoTokyoJapan

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