Abstract
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.
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References
U. Fano,Rev. Mod. Phys. 29:74 (1957).
D. Ter Haar,Rep. Progr. Phys. (Inst. Phys. and Phys. Soc., London)24:304 (1961).
E. Lehmann,Testing of Statistical Hypotheses (John Wiley and Sons, New York, 1959).
D. Middleton,An Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), Chaps. 18–23.
I. Selin,Detection Theory (Princeton University Press, Princeton, New Jersey, 1965).
J. C. Hancock and P. A. Wintz,Signal Detection Theory (McGraw-Hill Book Co., New York, 1966).
A. V. Balakrishnan (ed.),Communication Theory (McGraw-Hill Book Co., New York, 1968), Chaps. 3 and 4.
C. W. Helstrom,Statistical Theory of Signal Detection (Pergamon Press, Oxford, England, 1968), 2nd. ed.
H. Van Trees,Detection, Estimation, and Modulation Theory (John Wiley and Sons, New York, 1968), Vol. 1.
T. Bayes,Phil. Trans. Roy. Soc. (London) 53:370 (1763); reprinted inBiometrika 45:293 (1958).
J. Neyman and E. Pearson,Proc. Cambridge Phil. Soc. 29:492 (1933).
J. Neyman and E. Pearson,Phil. Trans. Roy. Soc. (London) A231:289 (1933).
C. W. Helstrom,Information and Control 10:254 (1967).
C. W. Helstrom,Intern. J. Theoret. Phys. 1:37 (1968).
C. W. Helstrom,Information and Control 13:156 (1968).
F. Riesz and B. Sz.-Nagy,Functional Analysis (F. Ungar Publishing Co., New York, 1955), p. 238.
W. H. Louisell,Radiation and Noise in Quantum Electronics (McGraw-Hill Book Co., New York, 1964), p. 242.
P. A. Bakut and S. S. Shchurov,Problemy Peredachi Informatsii 4(1):77 (1968).
R. J. Glauber,Phys. Rev. 131:2766 (1966).
G. Lachs,Phys. Rev. 138:B1012 (1965).
C. W. Helstrom,Trans. IEEE AES-5:562 (1969).
D. Middleton,Trans. IEEE IT-12:230 (1966).
G. Doetsch,Handbuch der Laplace-Transformation (BirkhÄuser Verlag, Basel and Stuttgart, 1955), Vol. 2, Chap. 3.
C. W. Helstrom,J. Opt. Soc. Am. 59:924 (1969).
A. A. Kuriksha,Problemy Peredachi Informatsii 3(1):42 (1967).
A. A. Kuriksha,Radiotekhn. i Elektron. 13:1790 (1968).
J. von Neumann,Mathematische Grundlagen der Quanteumechanik (Springer Verlag, Berlin, 1932), p. 130.
H. Cramér,Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), p. 473ff.
C. R. Rao,Bull. Calcutta Math. Soc. 37:81 (1945).
C. W. Helstrom,Phys. Letters 25A:101 (1967).
C. W. Helstrom,Trans. IEEE IT-14:234 (1968).
H. Margenau and R. N. Hill,Progr. Theoret. Phys. (Kyoto) 26:722 (1961).
L. Schmetterer,Mathematische Statistik (Springer Verlag, New York, 1966), p. 63.
B. Reiffen and H. Sherman,Proc. IEEE 51:1316 (1963).
C. W. Helstrom,Trans. IEEE IT-10:274 (1964).
J. W. Goodman,IEEE J. Quant. Elec. QE-1:180 (1965).
J. W. Goodman,Trans. IEEE AES-2:526 (1966).
P. A. Bakut, V. G. Vygon, A. A. Kuriksha, V. G. Repin, and G. P. Tartakovskii,Problemy Peredachi Informatsii 2(4):39 (1966).
V. L. Stefanyuk,Problemy Peredachi Informatsii 2(1):58 (1966).
P. A. Bakut,Radio Eng. Electron. (USSR) 11:551 (1966).
S. A. Ginzburg,Radio Eng. Electron. (USSR) 11:1972 (1966).
P. A. Bakut,Radio Eng. Electron. (USSR) 12:1 (1967).
I. Bar-David,Trans. IEEE IT-15:31 (1969).
T. E. Stern,Trans. IRE IT-6:435 (1960).
J. P. Gordon,Proc. IRE 50:1898 (1962).
B. E. Goodwin and L. P. Bolgiano, Jr.,Proc. IEEE 53:1745 (1965).
L. F. Jelsma and L. P. Bolgiano,IEEE Ann. Commun. Conv. Record 1965, 635.
L. B. Levitin,Problemy Peredachi Informatsii 1(3):71 (1965).
H. Takahasi,Advances in Communication Systems, A. V. Balakrishnan, ed. (Academic Press, New York, 1965), Vol. 1, p. 227.
D. S. Lebedev and L. B. Levitin,Information and Control 9:1 (1966).
V. V. Mityugov,Problemy Peredachi Informatsii 2(3):48 (1966).
R. L. Stratonovich,Problemy Peredachi Informatsii 2(1):45 (1966).
S. I. Borovitskii and V. V. Mityugov,Problemy Peredachi Informatsii 3(1):35 (1967).
G. Fillmore and G. Lachs,Trans. IEEE IT-15:465 (1969).
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This paper was prepared under grant NGR-05-009-079 from the National Aeronautics and Space Administration.
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Helstrom, C.W. Quantum detection and estimation theory. J Stat Phys 1, 231–252 (1969). https://doi.org/10.1007/BF01007479
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DOI: https://doi.org/10.1007/BF01007479