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Electronics and Communications in Japan (Part I Communications)

Construction and analysis of a mathematical model in quantum communication processes

Masanori Ohya, Noboru Watanabe, Regular Members

Published Feb 1, 1985• 33 citations
Publisher

Abstract

Abstract The communication of information by means of laser is an important technique. Therefore, it is necessary to construct a communication theory (quantum information theory) including quantum effects of laser action. Ohya, one of the authors, has studied the quantum communication channel and constructed a compound state for systematic formulation of the mutual information in quantum systems. In this paper, (1) we briefly review the mathematical formulation of quantum communication theory and (2) construct a physical model of the optical fiber communication process in order to examine the qualitative properties of the communication channel.

DOI
10.1002/ecja.4410680204

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Abstract The communication of information by means of laser is an important technique. Therefore, it is necessary to construct a communication theory (quantum information theory) including quantum effects of laser action. Ohya, one of the authors, has studied the quantum communication channel and constructed a compound state for systematic formulation of the mutual information in quantum systems. In this paper, (1) we briefly review the mathematical formulation of quantum communication theory and (2) construct a physical model of the optical fiber communication process in order to examine the qualitative properties of the communication channel.
Quantum mutual entropy and quantum capacity are rigorously defined by Ohya, and they are quite useful in the study of quantum communication processes [4, 7, 8, 9,10]. Mathematical models of optical communication processes are described by a quantum channel and optical states, and quantum capacity is one of the most important criteria to measure the efficiency of information transmission [4,7,8]. In actual optical communication, a laser beam is used for a signal, and it is denoted mathematically by a coherent state. Further, optical communication using a squeezed state, which is expected to be more efficient than that using a coherent state is proposed. In this paper, we define several quantum channels, that is, a squeezed channel and a coherent channel and so on. We compare them by calculating quantum capacity.
In several complicated systems, it is important to study (1) the dynamics of state change and (2) the complexity of states of systems. Information Dynamics (ID) introduced by Ohya is a general idea constructing a theory by synthesizing the research schemes of (1) and (2) under a frame of the ID. In ID, there are two types of complexities, that is, (a) a complexity of state describing system itself and (b) a transmitted complexity between two systems. Entropies of classical and quantum information theory are the example of the complexities of (a) and (b). In quantum information theory, Ohya introduced a compound state and defined Ohya mutual entropy based on quantum relative entropy of Umegaki in 1983, he extended it to general quantum systems by using the relative entropy of Araki and Uhlmann. One can discuss the coding theorems by means of the mean entropy and the mean mutual entropy defined by the dynamical entropy. In this paper, we will discuss about complexity of quantum communication processes including quantum correlations.
The quantum entropy introduced by von Neumann around 1932 describes the amount of information of the quantum state itself. It was extended by Ohya for C*-systems before Conne-Narnhoffer-Thirring (CNT) entropy. The quantum relative entropy was first defined by Umegaki for σ-finite von Neumann algebras and it was subsequently extended by Araki and Uhlmann for general von Neumann algebras and *-algebras, respectively. By introducing a new notion, the so-called compound state, in 1983 Ohya succeeded to construct the mutual entropy in a complete quantum mechanical system (i.e., input state, output state and channel are all quantum mechanical) describing the amount of information correctly transmitted through the quantum channel. In this paper, we briefly review Ohya's S-mixing entropy and the quantum mutual entropy for general quantum systems. Based on a concept of structure equivalent, we apply the general framework of quantum communication to the Gaussian communication processes.
The most protected and stable communication systems today are quantum channels of information transmission and processing. Thanks to the unique properties of photons as information elements, it becomes possible to monitor and analyze the state of information flows in communication or information transmission channels. Physical attributes such as spin, polarization, radiation frequency, phase synchronization, and the quantum entanglement effect can be tracked and interpreted online to improve the quality and reliability of information in computer systems. In order to effectively use information in support or decision support systems, it is necessary to carefully formalize the processes and indicators of quantum systems for the creation of information processing and transmission, for which information and mathematical models should be created that describe the state of the quantum communication channel (QCC).The information model should allow the convolution of the information space. The mathematical model must prove the processes of tracking the states of quantum information and provide a description of the phase state of indicators of the quantum environment. A lock in a closed space with established cause-and-effect relationships is equal to a system of clear logic.The authors summarize the experience of developing and implementing the method of simulated dynamic modeling of events in an abstract communication channel, which allows formalizing and classifying cause-and-effect relationships of quantum carriers in the analyzed channels. It is proposed to use a unified neural network for the organization of SPPR in quantum-mechanical information transmission systems. Such a network could provide an automatic intelligent system state analysis mode. Such an analysis makes it possible to classify the aggregates of current system parameters to the level of diagnostics of the state of information flows and conclusions based on such diagnostics with the support of decision-making about the quality and reliability of the transmitted information. Such a system, working in OLAP (Online analytical processing) mode, could automatically manage the process of generation and transmission of information, reacting without human intervention to emerging critical errors or attempts at unauthorized system hacking. The observer effect leads to the fact that an attempt to measure the state of a photon inevitably causes an almost instantaneous change in this state. Attempting to parallelize a photon has the same consequences. This cannot be unnoticeable during further authorized acceptance of information. The analyzed quantum communication channel (QCM) consists of a set of technological elements distributed in space. The channel works in its own time, which is formed by clock pulses and creates a flow of information.
In communication processes, a channel has an activity to communicate information of input system to output system. The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983, and he extended it to general quantum systems by means of the relative entropy of Araki and Uhlmann. The capacity shows the ability of the information transmission of the channel, which is used as a measure for construction of channels. The fully quantum capacity is formulated by taking the supremum of Ohya mutual entropy with respect to a certain subset of the initial state space. One of the most important theorems in quantum communication theory is coding theorem. The quantum coding theorems are discussed by using the mutual entropy — type measures introduced in the studies of quantum information.In this paper, we compare with these mutual entropy‐type measures in order to obtain most suitable one for discussing the quantum coding theorem. We concretely calculate the quantum capacity for quantum channels.