Classical dynamical entropy is an important tool to analyse the efficiency of infor- mation transmission in communication processes.Here we report new formulations of quantum dynamical entropy.
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.
This paper introduces a novel approach to information theory, leveraging quantum entanglement entropy as a fundamental measure of information. We propose that entanglement entropy can effectively quantify the efficiency of information transmission and provides a theoretical foundation that surpasses the limitations of classical information theory. The core of this work lies in the integration of entanglement entropy with Shannon entropy, establishing a new framework for analyzing information transfer. We explore scenarios where classical information theory struggles to adequately describe the limits of communication due to quantum mechanical effects, specifically entanglement. The paper aims to provide a theoretical underpinning for quantum communication protocols and potentially inform the development of quantum computation. We define key terms and formulate core concepts, focusing on the relationship between entanglement entropy, information content, and the constraints imposed by quantum mechanics. The central argument is that by considering the inherent interconnectedness of entangled systems, we can gain a deeper understanding of information processing at the most fundamental level. ---
Abstract This book deals with the basic concept of information, its importance, and the similarities and differences between classical and quantum information. First, classical information theory and its application to communication are discussed. Quantum mechanics and communication based on the laws of quantum mechanics are then considered. Quantum entanglement, a key resource in quantum communication, is analysed both from the fundamental perspective and from the perspective of its information-processing capability. This book also introduces quantum computation (in the context of the search and factorisation algorithms) and the basics of quantum error correction. Throughout this book, the connections between information theory, thermodynamics, and (quantum) physics are emphasised and discussed. Other topics covered in the book are data compression, related measures of information such as relative entropy, joint entropy, conditional entropy, and mutual information, along with the capacity of a noisy channel.
In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy. Entropy is an essential notion in physics and information theory. Motivated by the study of disorder for the positions and velocities of gas molecules, the notion of entropy was first introduced mathematically by Boltzmann near the end of the 19th Century and gives rise to the second law of thermodynamics. Almost eighty years after Boltzmann, Shannon became the father of the new field of information theory when he produced his groundbreaking works where he used entropy as a measure of information transfer between two sources. In the last two years of the 1950’s, Kolmogorov and Sinai extended the notions of Boltzmann to a dynamical entropy. The Kolmogorov-Sinai dynamical entropy gives a measure for the disorder of a system of particles (e.g. gas molecules) averaged over time, quantifying the uncertainty in the dynamics of a system. The advent of quantum mechanics and its pervasiveness in nature has required the development of non-commutative generalizations of dynamics and dynamical entropy to the quantum regime. Many of each have been proposed. In particular, we recall the definitions of quantum random walks, dynamical systems and Markov chains. We motivate each generalization by relating to its classical counterpart. Quantum dynamical entropy (QDE) is a generalization of the Kolmogorov-Sinai dynamical entropy to quantum mechanics. There have been numerous definitions for QDE beginning with that of Connes, Narnhofer and Thirring in 1987. We focus on the semi- classical approach given by Słomczyński and Życzkowski in 1994 and the quantum Markov chain approach which started with Accardi, Ohya and Watanabe in 1997. Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Słomczyński and Życzkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Lüders-von Neumann instruments and partitions. In 1948, Shannon proved the Source Coding Theorem which gives upper and lower bounds on the minimal expected codeword length in terms of the entropy of a ran- dom variable. This theorem can be leveraged to give the minimal expected average codeword length for a string of symbols in terms of entropy rate, which can be in- terpreted as the dynamical entropy of a stochastic process. In 1994, Schumacher defined indeterminate-length quantum codes and proved a Quantum Source Coding Theorem. We introduce the notion of stochastic ensembles of pure states and give a novel representation in terms of quantum Markov chains. Moreover, this representa- tion allows us to extend the Quantum Source Coding Theorem, giving the minimal expected average codeword length of an indeterminate-length quantum code in terms of quantum dynamical entropy. This manuscript includes joint work with Dr. George Androulakis (Mathematics, University of South Carolina).
The classical entropies were formulated by means of the classical probability theory. The essence of the classical probability theory corresponds to the use of c‐number with commutative property. On the base of the classical probability theory, the classical information theory can be discussed by using the classical entropies. On the other hand, the principle of the quantum phenomenon seems to be having non‐commutative behavior. It leads to the use of the q‐operator with non‐commutative property to introduce the quantum entropies based on the quantum probability theory. In order to discuss the efficiency of the information communication processes, a measure of complexity of initial state itself and a measure of transmitted complexity through communication channels are necessary. Shannon’s entropy and the mutual entropy represent these measures of complexities in classical communication system, respectively. Von Neumann entropy and Ohya mutual entropy relate to these measures of complexities in quantum communication system, respectively. Recently, several mutual entropy type measures (Lindblad—Nielsen entropy and Coherent entropy) were proposed by using an entropy exchange with respect to a channel and an initial state. In this paper, we will show the most suitable one for discuss the efficiency of information transmission for quantum processes. In classical information theory, one of the most important tools to analyze communication processes are the classical coding theorems, which was investigated through the classical dynamical entropy. The classical dynamical entropy was discussed in [1] with respect to a measure preserving transformation defined on a message space for finite partitions of the measure space. In quantum information theory, Emch [2] and Conne, Stormer [3] first studied a complexity of quantum dynamical processes. After that, Ohya introduced a S‐ mixing entropy in general quantum systems, which was natural extension of the von Neumann entropy. Using the S—mixing entropy, Ohya defined a mean entropy and a mean mutual entropy for the quantum dynamical systems. Conne, Narnhoffer and Thirring introduced a dynamical entropy (CNT entropy) [4] and several researchers discussed the CNT entropy [5,6,7,8]. Alicki and Fannes defined a dynamical entropy (AF entropy) [9] using a finite operational partitions of unity. In 1995, Voiculescu proposed a dynamical entropy (dynamical approximation entropy) [10] in C* and W* algebras. Through the quantum Markov process, Accardi, Ohya and Watanabe defined a dynamical entropy (AOW entropy) [11,12] in 1997. Base on transition expectations and liftings in the sense of Accardi and Ohya [13], Kossakowski, Ohya and Watanabe introduced a dynamical entropy (KOW entropy) [14] with respect to completely positive maps in 1999. In this paper, we will discuss the complexity of the quantum dynamical process to calculate the generalized AOW entropy given by KOW entropy.