Journal of Statistical Physics

, Volume 1, Issue 2, pp 231–252 | Cite as

Quantum detection and estimation theory

  • Carl W. Helstrom


A review. Quantum detection theory is a reformulation, in quantum-mechanical terms, of statistical decision theory as applied to the detection of signals in random noise. Density operators take the place of the probability density functions of conventional statistics. The optimum procedure for choosing between two hypotheses, and an approximate procedure valid at small signal-to-noise ratios and called threshold detection, are presented. Quantum estimation theory seeks best estimators of parameters of a density operator. A quantum counterpart of the Cramér-Rao inequality of conventional statistics sets a lower bound to the mean-square errors of such estimates. Applications at present are primarily to the detection and estimation of signals of optical frequencies in the presence of thermal radiation.

Key words

signal detection detection theory parameter estimation statistical estimation estimation theory quantum theory decision theory hypothesis testing statistical decisions 


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  1. 1.
    U. Fano,Rev. Mod. Phys. 29:74 (1957).Google Scholar
  2. 2.
    D. Ter Haar,Rep. Progr. Phys. (Inst. Phys. and Phys. Soc., London)24:304 (1961).Google Scholar
  3. 3.
    E. Lehmann,Testing of Statistical Hypotheses (John Wiley and Sons, New York, 1959).Google Scholar
  4. 4.
    D. Middleton,An Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), Chaps. 18–23.Google Scholar
  5. 5.
    I. Selin,Detection Theory (Princeton University Press, Princeton, New Jersey, 1965).Google Scholar
  6. 6.
    J. C. Hancock and P. A. Wintz,Signal Detection Theory (McGraw-Hill Book Co., New York, 1966).Google Scholar
  7. 7.
    A. V. Balakrishnan (ed.),Communication Theory (McGraw-Hill Book Co., New York, 1968), Chaps. 3 and 4.Google Scholar
  8. 8.
    C. W. Helstrom,Statistical Theory of Signal Detection (Pergamon Press, Oxford, England, 1968), 2nd. ed.Google Scholar
  9. 9.
    H. Van Trees,Detection, Estimation, and Modulation Theory (John Wiley and Sons, New York, 1968), Vol. 1.Google Scholar
  10. 10.
    T. Bayes,Phil. Trans. Roy. Soc. (London) 53:370 (1763); reprinted inBiometrika 45:293 (1958).Google Scholar
  11. 11.
    J. Neyman and E. Pearson,Proc. Cambridge Phil. Soc. 29:492 (1933).Google Scholar
  12. 12.
    J. Neyman and E. Pearson,Phil. Trans. Roy. Soc. (London) A231:289 (1933).Google Scholar
  13. 13.
    C. W. Helstrom,Information and Control 10:254 (1967).Google Scholar
  14. 14.
    C. W. Helstrom,Intern. J. Theoret. Phys. 1:37 (1968).Google Scholar
  15. 15.
    C. W. Helstrom,Information and Control 13:156 (1968).Google Scholar
  16. 16.
    F. Riesz and B. Sz.-Nagy,Functional Analysis (F. Ungar Publishing Co., New York, 1955), p. 238.Google Scholar
  17. 17.
    W. H. Louisell,Radiation and Noise in Quantum Electronics (McGraw-Hill Book Co., New York, 1964), p. 242.Google Scholar
  18. 18.
    P. A. Bakut and S. S. Shchurov,Problemy Peredachi Informatsii 4(1):77 (1968).Google Scholar
  19. 19.
    R. J. Glauber,Phys. Rev. 131:2766 (1966).Google Scholar
  20. 20.
    G. Lachs,Phys. Rev. 138:B1012 (1965).Google Scholar
  21. 21.
    C. W. Helstrom,Trans. IEEE AES-5:562 (1969).Google Scholar
  22. 22.
    D. Middleton,Trans. IEEE IT-12:230 (1966).Google Scholar
  23. 23.
    G. Doetsch,Handbuch der Laplace-Transformation (BirkhÄuser Verlag, Basel and Stuttgart, 1955), Vol. 2, Chap. 3.Google Scholar
  24. 24.
    C. W. Helstrom,J. Opt. Soc. Am. 59:924 (1969).Google Scholar
  25. 25.
    A. A. Kuriksha,Problemy Peredachi Informatsii 3(1):42 (1967).Google Scholar
  26. 26.
    A. A. Kuriksha,Radiotekhn. i Elektron. 13:1790 (1968).Google Scholar
  27. 27.
    J. von Neumann,Mathematische Grundlagen der Quanteumechanik (Springer Verlag, Berlin, 1932), p. 130.Google Scholar
  28. 28.
    H. Cramér,Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), p. 473ff.Google Scholar
  29. 29.
    C. R. Rao,Bull. Calcutta Math. Soc. 37:81 (1945).Google Scholar
  30. 30.
    C. W. Helstrom,Phys. Letters 25A:101 (1967).Google Scholar
  31. 31.
    C. W. Helstrom,Trans. IEEE IT-14:234 (1968).Google Scholar
  32. 32.
    H. Margenau and R. N. Hill,Progr. Theoret. Phys. (Kyoto) 26:722 (1961).Google Scholar
  33. 33.
    L. Schmetterer,Mathematische Statistik (Springer Verlag, New York, 1966), p. 63.Google Scholar
  34. 34.
    B. Reiffen and H. Sherman,Proc. IEEE 51:1316 (1963).Google Scholar
  35. 35.
    C. W. Helstrom,Trans. IEEE IT-10:274 (1964).Google Scholar
  36. 36.
    J. W. Goodman,IEEE J. Quant. Elec. QE-1:180 (1965).Google Scholar
  37. 37.
    J. W. Goodman,Trans. IEEE AES-2:526 (1966).Google Scholar
  38. 38.
    P. A. Bakut, V. G. Vygon, A. A. Kuriksha, V. G. Repin, and G. P. Tartakovskii,Problemy Peredachi Informatsii 2(4):39 (1966).Google Scholar
  39. 39.
    V. L. Stefanyuk,Problemy Peredachi Informatsii 2(1):58 (1966).Google Scholar
  40. 40.
    P. A. Bakut,Radio Eng. Electron. (USSR) 11:551 (1966).Google Scholar
  41. 41.
    S. A. Ginzburg,Radio Eng. Electron. (USSR) 11:1972 (1966).Google Scholar
  42. 42.
    P. A. Bakut,Radio Eng. Electron. (USSR) 12:1 (1967).Google Scholar
  43. 43.
    I. Bar-David,Trans. IEEE IT-15:31 (1969).Google Scholar
  44. 44.
    T. E. Stern,Trans. IRE IT-6:435 (1960).Google Scholar
  45. 45.
    J. P. Gordon,Proc. IRE 50:1898 (1962).Google Scholar
  46. 46.
    B. E. Goodwin and L. P. Bolgiano, Jr.,Proc. IEEE 53:1745 (1965).Google Scholar
  47. 47.
    L. F. Jelsma and L. P. Bolgiano,IEEE Ann. Commun. Conv. Record 1965, 635.Google Scholar
  48. 48.
    L. B. Levitin,Problemy Peredachi Informatsii 1(3):71 (1965).Google Scholar
  49. 49.
    H. Takahasi,Advances in Communication Systems, A. V. Balakrishnan, ed. (Academic Press, New York, 1965), Vol. 1, p. 227.Google Scholar
  50. 50.
    D. S. Lebedev and L. B. Levitin,Information and Control 9:1 (1966).Google Scholar
  51. 51.
    V. V. Mityugov,Problemy Peredachi Informatsii 2(3):48 (1966).Google Scholar
  52. 52.
    R. L. Stratonovich,Problemy Peredachi Informatsii 2(1):45 (1966).Google Scholar
  53. 53.
    S. I. Borovitskii and V. V. Mityugov,Problemy Peredachi Informatsii 3(1):35 (1967).Google Scholar
  54. 54.
    G. Fillmore and G. Lachs,Trans. IEEE IT-15:465 (1969).Google Scholar

Copyright information

© Plenum Publishing Corporation 1969

Authors and Affiliations

  • Carl W. Helstrom
    • 1
  1. 1.Department of Applied ElectrophysicsUniversity of California at San DiegoLa Jolla

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