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 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.
von Neumann entropy represents the amount of information in the quantum state, and this was extended by Ohya for general quantum systems [10]. Umegaki first defined the quantum relative entropy for -finite von Neumann algebras, which was extended by Araki, and Uhlmann, for general von Neumann algebras and *-algebras, respectively. In 1983 Ohya introduced the quantum mutual entropy by using compound states; this describes the amount of information correctly transmitted through the quantum channel, which was also extended by Ohya for general quantum systems. In this paper, we briefly explain Ohya?s -mixing entropy and the quantum mutual entropy for general quantum systems. By using structure equivalent class, we will introduce entropy type functionals based on quantum information theory to improve treatment for the Gaussian communication process.
In 1989, Ohya propose a new concept, so-called Information Dynamics (ID), to investigate complex systems according to two kinds of view points.One is the dynamics of state change and another is measure of complexity.In ID, two complexities C S and T S are introduced.C S is a measure for complexity of system itself, and T S is a measure for dynamical change of states, which is called a transmitted complexity.An example of these complexities of ID is entropy for information transmission processes.The study of complexity is strongly related to the study of entropy theory for classical and quantum systems.The quantum entropy was introduced by von Neumann around 1932, which describes the amount of information of the quantum state itself.It was extended by Ohya for C*-systems before CNT entropy.The quantum relative entropy was first defined by Umegaki for σ-finite von Neumann algebras, which was 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 formulate 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 the entropic complexities for classical and quantum systems.We introduce some complexities by means of entropy functionals in order to treat the transmission processes consistently.We apply the general frames of quantum communication to the Gaussian communication processes.Finally, we discuss about a construction of compound states including quantum correlations.
The quantum entropy was introduced by von Neumann around 1932, which describes the amont of information of the quantum state itself. It was extended by Ohya for C*‐systems. The quantum relative entropy was first defined by Umegaki for σ ‐finite von Neumann algebras, which denotes a certain difference between two states. It was extended by Araki and Uhlmann for general von Neumann algebras and *‐algebras, respectively. The development of communication theory is closely connected with study of entropy theory. In order to discuss the commuinication processes, we need at least two systems, such as input and output systems. The signal of the input system is carried through a physical device, which is called a channel. The mathematical representation of the channel is a mapping from the input state space to the output state space. The mutual entropy (information) defined by using the relative entropy between the product state consist of the input state and the output state and the compound state given by the communication process, expresses the amount of information correctly transmitted through the channel from the input system to the output system. In classical communication theory, the mutual entropy was formulated by using the joint probability distribution between the input system and the output system. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. In fully quantum system, there does not exist the joint probability distribution in general. Instead of the joint probability distribution, Ohya invented the quantum (Ohya) compound state, and he introduced the fully quantum mutual entropy (information), which is called Ohya mutual entropy, for quantum input and output systems, describes the amount of information correctly sent from the quantum input system to the quantum output system through the quantum channel. It was extended by Ohya for general quantum systems. Recently Shor and Bennet et al took the coherent entropy and defined the mutual type entropy to discuss a sort of coding theorem for communication processes.In this paper, we concretely calculate these mutual entropy‐type measures for the attenuation channel in order to assay these abilities.
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.
In classical information theory, the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov–Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes–Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy were proposed [10,[Formula: see text]3,[Formula: see text]8,[Formula: see text]39,[Formula: see text]15,[Formula: see text]44,[Formula: see text]9,[Formula: see text]27,[Formula: see text]28,[Formula: see text]2,[Formula: see text]19,[Formula: see text]45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S-mixing entropy by Ohya [26,[Formula: see text]27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S-mixing entropy. In this paper, based on research into the dynamical entropy and mean mutual entropy of quantum systems, this study investigates the characteristics of quantum channels that exhibit entanglement, a property unique to quantum systems, and investigates the behaviour of quantum entropy for compound quantum channels by changing the perspective from information transmission between input and output to a new perspective of information transmission from the initial state to the final state.
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.