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IEEE International Conference on Communication Systems (ICCS)

The distance between two signals on the quantum and the classical communication theory

Y. Sakamoto, Masao Osaki, Osamu Hirota, Masaki Nakagawa

Published Jan 1, 1992• 0 citations
Publisher

Abstract

In order to build a new standard of evaluation for the distinction of signals, the authors propose the orthogonality on quasi-phase space, which corresponds to the inner product on the Hilbert space. Hence the quasi-phase space is the generalized signal space where some concepts of quantum theory work. This orthogonality has great advantages of making possible to define the distance between two signals without a precise representation of the quantum state and providing a very convenient method of analysis of quantum communication systems. Moreover, since the mathematical structure of the orthogonality is similar to that of the Bhattacharyya distance in the classical theory, the authors discuss whether it is possible or not to apply the orthogonality to the conventional communication system.>

DOI
10.1109/iccs.1992.254985

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In order to build a new standard of evaluation for the distinction of signals, the authors propose the orthogonality on quasi-phase space, which corresponds to the inner product on the Hilbert space. Hence the quasi-phase space is the generalized signal space where some concepts of quantum theory work. This orthogonality has great advantages of making possible to define the distance between two signals without a precise representation of the quantum state and providing a very convenient method of analysis of quantum communication systems. Moreover, since the mathematical structure of the orthogonality is similar to that of the Bhattacharyya distance in the classical theory, the authors discuss whether it is possible or not to apply the orthogonality to the conventional communication system.>
A method of defining structural information in Hilbert space is described. Instead of commencing an information theory analysis with a simple signal and progressing to an n-dimensional information space, the reverse procedure can be followed: the signal is defined completely in both real and imaginary spaces, and is then represented as a dimensionless number in Hilbert space. Thus, a signal can be analyzed into a Hilbert space measure of information. This measure has similarities to the relativity of states of Everett's (1957) reformulation of quantum mechanics involving observer-observed interaction. Whereas Everett's theory deals with the totality of all possible ways in which a state function can be decomposed into the sum of products of state functions for subsystems of the overall system, the theory presented here deals with the totality of possible ways in which a Hilbert space measure of signal state can be decomposed by signal definition into reciprocally related measures of the state for subsystems of the overall system. In the overall system presented here, a signal is given a concise representation either as a quadratic form on a Hilbert space, or as a Fourier coefficients in isomorphic function space. The total information content of a signal qualified as a dimensionless number in Hilbert space may be given a representation in familiar units by the use of such methods. A primary postulate of quantum mechanics is obeyed in that all physically relevant information about a system is derivable from the knowledge of the state function of the system. Thus, a signal is defined firstly as an information measure, and secondly in the familiar units of frequency, bandwidth, midperiod, and duration.
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Some general principles There are some important differences between quantum and classical reasoning which reflect the different mathematical structure of the two theories. The most precise classical description of a mechanical system is provided by a point in the classical phase space, while the most precise quantum description is a ray or onedimensional subspace of the Hilbert space. This in itself is not an important difference. What is more significant is the fact that two distinct points in a classical phase space represent mutually exclusive properties of the physical system: if one is a true description of the sytem, the other must be false. In quantum theory, on the other hand, properties are mutually exclusive in this sense only if the corresponding projectors are mutually orthogonal. Distinct rays in the Hilbert space need not be orthogonal to each other, and when they are not orthogonal, they do not correspond to mutually exclusive properties. As explained in Sec. 4.6, if the projectors corresponding to the two properties do not commute with one another, and are thus not orthogonal, the properties are (mutually) incompatible. The relationship of incompatibility means that the properties cannot be logically compared, a situation which does not arise in classical physics. The existence of this nonclassical relationship of incompatibility is a direct consequence of assuming (following von Neumann) that the negation of a property corresponds to the orthogonal complement of the corresponding subspace of the Hilbert space; see the discussion in Sec. 4.6.
The phase space representation is used in the classical mechanics of particles but is not appropriate for describing interference phenomena that occur in classical field theory such as classical optics. The methods of classical optics and quantum mechanics are amazingly similar. That is principally because both use Hilbert spaces to represent states. The only difference is in the use of non-commuting operators to represent observables in quantum mechanics. The quantum theory of light predicts perfect anti-coincidence for single detections on the two sides of a beam splitter, whereas classical light, however weak, would be divided on a beam splitter and produce a minimum degree of correlated or coincidence counts. In mathematics the Hermite polynomials are classical orthogonal polynomial sequences that arise in probability and numerical analysis as Gaussian quadrature. They are also used in systems theory in connection with non-linear operations on Gaussian noise and other sciences.
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