Adaptive Dynamics and Its Applications to Chaos and NPC Problem Quantum Algorithm for SAT Problem and Quantum Mutual Entropy New Quantum Algorithm for Studying NP-Complete Problems Entanglement, Quantum Entropy and Mutual Information Fundamentals of Quantum Mutual Entropy and Capacity Compound Channels, Transition Expectations, and Liftings Complexities and Their Applications to Characterization of Chaos Complexity, Fractal Dimension for Quantum States Information Dynamics and Its Applications to Optical Communication Processes Information Theoretical Treatments of Genes Some Aspects of Quantum Information Theory and Their Applications to Irreversible Processes On Compound State and Mutual Information in Quantum Information Theory Note on Quantum Probability Quantum Ergodic Channels in Operator Algebras On Open System Dynamics -- An Operator Algebraic Study Dynamical Process in Linear Response Theory Stability of Weiss Ising Model and other papers.
Adaptive Dynamics and Its Applications to Chaos and NPC Problem Quantum Algorithm for SAT Problem and Quantum Mutual Entropy New Quantum Algorithm for Studying NP-Complete Problems Entanglement, Quantum Entropy and Mutual Information Fundamentals of Quantum Mutual Entropy and Capacity Compound Channels, Transition Expectations, and Liftings Complexities and Their Applications to Characterization of Chaos Complexity, Fractal Dimension for Quantum States Information Dynamics and Its Applications to Optical Communication Processes Information Theoretical Treatments of Genes Some Aspects of Quantum Information Theory and Their Applications to Irreversible Processes On Compound State and Mutual Information in Quantum Information Theory Note on Quantum Probability Quantum Ergodic Channels in Operator Algebras On Open System Dynamics -- An Operator Algebraic Study Dynamical Process in Linear Response Theory Stability of Weiss Ising Model and other papers.
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.
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.
In quantum information theory, Emch, Conne, and Stormer were the first who studied the complexity of quantum dynamical processes. After that, Ohya introduced the [Formula: see text]-mixing entropy for general quantum systems and he defined the mean entropy and the mean mutual entropy for quantum dynamical systems based on the [Formula: see text]-mixing entropy. Conne, Narnhoffer and Thirring introduced the dynamical entropy (CNT entropy) and several researchers discussed this concept. Alicki and Fannes defined a different dynamical entropy — AF entropy. In 1995, Voiculescu proposed the dynamical approximation entropy. Accardi, Ohya and Watanabe defined yet another dynamical entropy (AOW entropy) through a quantum Markov process in 1997. In 1999, Kossakowski, Ohya and Watanabe introduced the dynamical entropy (KOW entropy) with respect to completely positive maps. In this paper, we discuss the complexity of quantum dynamical processes to calculate the dynamical entropy for noisy optical channels. In order to discuss the efficiency of information communication processes, a measure of complexity of initial state itself and a measure of transmitted complexity through communication channels are necessary. Quantum entropies were formulated on the basis of the quantum probability theory. In quantum communication systems, von Neumann entropy and Ohya mutual entropy relate to these measures of complexities, respectively. Recently, several mutual entropy type measures (Lindblad-Nielsen entropy and coherent entropy) were defined making use of entropy exchange with respect to a channel and initial state. In this paper, we show which of the measures is the most suitable one for discussing the efficiency of information transmission for quantum 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.
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.
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.