Information Dynamics and Its Application to Gaussian Communication Processes

Abstract

The ”dynamics of systems” is described by the state change. One of the essential characters of a state is expressed by complexity, such as entropy, an important concept in information theory. Information dynamics introduced in [4] is for the study of the state change together with such complexities, being a synthesis of two concepts; dynamics and complexity. There are two complexities in information dynamics; (i) the first complexity is one for a state itself, (ii) the second complexity (transmitted complexity) is determined for a state and a channel.

In this paper, we apply the general frames of information dynamics to Gaussian communication processes.

Gaussian communication processes have been studied by Baker [1] et al. They used the classical mutual entropy I(μ; Г) to discuss Gaussian communication processes. Their discussion has two defects [5]: (1) If we take the differential entropy as information for an input state, then this entropy often becomes less than the mutual entropy I(μ; Г); (2) If we take Shannon’s definition for the entropy S(p), then it is always infinite for any Gaussian measure μ. These points are not well-matched to the thought of Shannon [7].

In order to avoid these defects and discuss Gaussian communication processes consistently, we introduced [5] two entropy functionals based on the quantum von Neumann entropy [8] and the quantum mutual entropy (information) introduced in [2]. In this paper, we show that two functionals satisfy the conditions of complexities in information dynamics.

In §1, we explain what information dynamics is.

In §2, we briefly review the conventional treatment (Baker et al.) of Gaussian communication processes.

In §3, we explain the theory of quantum entropies for density operators.

In §4, based on quantum entropies, our entropy functional for an input state and a mutual entropy functional for an input state and a Gaussian channel satisfy the conditions of complexities of information dynamics.