The classical dynamical mutual entropy measures the average of information content going through a channel. Classical Markovian sources are important in communication theory since they constitute reasonable models for languages. In this paper, we define the quantum dynamical mutual entropy through quantum Markov chains, and we calculate it for several simple models.
The classical dynamical mutual entropy measures the average of information content going through a channel. Classical Markovian sources are important in communication theory since they constitute reasonable models for languages. In this paper, we define the quantum dynamical mutual entropy through quantum Markov chains, and we calculate it for several simple models.
In classical information theory, one of 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 are done [10, 3, 8, 39, 15, 44, 9, 27, 28, 2, 19, 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, 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, we discuss the efficiency of information transmission to calculate the mean mutual entropy with respect to the modulated initial states and the connected channel for the quantum dynamical systems.
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 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).