Abstract
In order to discuss communication processes consistently for a Gaussian input with a Gaussian channel on an infinite dimensional Hilbert space, we introduce the entropy functional of an input source and the mutual entropy functional for a Gaussian channel and show a fundamental inequality for communication processes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. R. Baker, Capacity of the Gaussian channel without feedback. Inform. and Control,37 (1978), 70–89.
I. M. Gelfand and A. M. Yaglom, Calculation of the amount of information about a random function contained in another such function. Amer. Math. Soc. Transl.12 (1959), 199–246.
C. R. Helstrom, J. W. S. Liu and J. P. Gordon, Quantum mechanical communication theory. Proc. IEEE,58 (1970), 1578–1598.
S. Ihara, On the capacity of the discrete time Gaussian channel with feedback. Trans. Eighth Prague Conf., Vol. C, Czechoslovak Acad. Sci., 1979, 175–186.
R. S. Ingarden, Quantum information theory, Rep. Math. Phys.10 (1976), 43–73.
S. Kullback and R. A. Leibler, On information and suffciency. Ann. Math. Statistics,21 (1953), 79–86.
H. H. Kuo, Gaussian Measures in Banach Spaces, Springer, Berlin, 1975.
M. Ohya, Quantum ergodic channels in operator algebras. J. Math. Anal. Appl.,84 (1981), 318–328.
M. Ohya, On compound state and mutual information in quantum information theory. IEEE Trans. Inform. Theory,29 (1983), 770–774.
M. Ohya, Note on quantum probability. Lett. Nuovo Cimento,38 (1983), 402–404.
M. Ohya, Entropy transmission inC *-dynamical systems. J. Math. Anal. Appl.,100 (1984), 222–235.
M. Ohya, State change and entropies in quantum dynamical systems. Lecture Note in Math.,1136, Springer, 1985, 397–408.
M. Ohya and N. Watanabe, Construction and analysis of a mathematical model in quantum communication processes. IECE of Japan, J67-A, No. 6 (1984), 548–552.
M. S. Pinsker, Informations and Information Stability of Random Variable and Processes. Holden-Day, Inc., 1964.
C. R. Rao and V. S. Varadarajan, Discrimination of Gaussian processes. Sankhya, Ser. A,25 (1963), 303–330.
C. Shannon, A mathematical theory of communication. Bell System Tech J.,27 (1948), 379–423, 623–656.
R. Schatten, Norm Ideals of Completely Continous Operators. Springer-Verlag, Berlin/New York, 1960.
A. V. Skorohod, Integration in Hilbert Space. springer-Verlag, Berlin, New York, 1974.
H. Takahashi, Information theory of quantum mechanical channels. Advances in Communication Systems. Vol. 1. Academic Press, 1966, 227–310.
M. Takesaki, Theory of Operator Algebra I, Springer-Verlag, New York, 1981.
H. Umegaki, Conditional expectation in an operator algebra. Tôhoku Math. J.,6 (1954), 177–181.
H. Umegaki, Conditional expectation in an operator algebra, III Kõdai Math. Sem. Rep.,11 (1959), 51–64.
H. Umegaki, Conditional expectation in an operator algebra, IV (entropy and information). Kõdai Math. Sem. Rep.,14 (1962), 59–85.
H. Umegaki, General treatment of alphabet-message space and integral representation of entropy. Kõdai Math. Sem. Rep.,16 (1964), 18–26.
J. von Neumann, Die Mathematischen Grundlagen der Quantemechnik, Springer, Berlin, 1932.
K. Yanagi, On some properties of Gaussian channels. J. Math. Anal. Appl.,88 (1982), 364–377.
Author information
Authors and Affiliations
Additional information
(Dedicated to Professor H. Umegaki on his 60th birthday)
About this article
Cite this article
Ohya, M., Watanabe, N. A new treatment of communication processes with Gaussian channels. Japan J. Appl. Math. 3, 197–206 (1986). https://doi.org/10.1007/BF03167097
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03167097