Abstract
We calculate the Fisher information matrix of Husimi distribution in the Fock–Bargmann representation. It turns out that the Fisher information of the position and that of the momentum move in opposite directions, and that a weighted trace of the Fisher information matrix is a constant independent of the Husimi distribution. This may be interpreted as a kind of uncertainty relation (in the spirit of Heisenberg uncertainty principle) from the statistical inference point of view.
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REFERENCES
S. Amari, Information geometry, in Geometry and Nature, H. Nencka and J. P. Bourguignon, eds. (American Mathematical Society, Providence, 1994), pp. 81-95.
V. Bargmann, On a Hilbert space of holomorphic functions and an associated integral transform, Commun. Pure Appl. Math. 14:187-214 (1961).
E. R. Caianielle, Quantum and other physics as system theory, La Rivista del Nuovo Cimento 15:1-65 (1992).
E. Carlen, Some integral identities and inequalities for entire functions and their applications to coherent state transform, J. Funct. Anal. 97:231-249 (1991).
E. Carlen, Superadditivity of Fisher's information and logarithmic Sobolev inequalities, J. Funct. Anal. 101:194-211 (1991).
R. A. Fisher, Theory of statistical estimation, Proc. Camb. Phil. Soc. 22:700-725 (1925).
B. R. Frieden, Physics from Fisher Information, a Unification (Cambridge University Press, 1998).
L. Gross, Logarithmic Sobolev inequality, Amer. J. Math. 97:1061-1083 (1975).
M. Hillery, R. E. O'Connel, M. O. Scully, and E. P. Wigner, Distribution functions in physics: fundamentals, Phys. Rep. 106:121-167 (1984).
K. Husimi, Some formal properties of the density matrix, Proc. Phys. Math. Soc. Japan 22:264-283 (1940).
E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106:620-630 (1957).
H. W. Lee, Theory and applications of the quantum phase-space distribution functions, Phys. Rep. 259:147-211 (1995).
C. R. Rao, Information and accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc. 37:81-91 (1945).
A. Wehrl, On the relation between classical and quantum entropy, Rep. Math. Phys. 16:353-358 (1979).
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Luo, S. Fisher Information Matrix of Husimi Distribution. Journal of Statistical Physics 102, 1417–1428 (2001). https://doi.org/10.1023/A:1004856832310
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DOI: https://doi.org/10.1023/A:1004856832310