We compute the dynamical entropy in the sense of Connes, Narnhofer, and Thirring of generalized quantum Markov chain over infinite dimensional algebras. For the case in which the transition expectation is defined by a set of conditional density amplitudes, we show that the dynamical entropy is equal to the mean entropy. Thus we extend the result of Park [Lett. Math. Phys. 32, 63–74 (1994)] to non-AF type quantum Markov chains. We employ the main method developed in Park [Lett. Math. Phys. 32, 63–74 (1994)] with necessary modifications.
We compute the dynamical entropy in the sense of Connes, Narnhofer, and Thirring of generalized quantum Markov chain over infinite dimensional algebras. For the case in which the transition expectation is defined by a set of conditional density amplitudes, we show that the dynamical entropy is equal to the mean entropy. Thus we extend the result of Park [Lett. Math. Phys. 32, 63–74 (1994)] to non-AF type quantum Markov chains. We employ the main method developed in Park [Lett. Math. Phys. 32, 63–74 (1994)] with necessary modifications.
Several quantum dynamical entropies have been proposed that extend the classical Kolmogorov–Sinai (dynamical) entropy. The same scenario appears in relation to the extension of algorithmic complexity theory to the quantum realm. A theorem of Brudno establishes that the complexity per unit time step along typical trajectories of a classical ergodic system equals the KS-entropy. In the following, we establish a similar relation between the Connes–Narnhofer–Thirring quantum dynamical entropy for the shift on quantum spin chains and the Gács algorithmic entropy. We further provide, for the same system, a weaker linkage between the latter algorithmic complexity and a different quantum dynamical entropy proposed by Alicki and Fannes.
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).
In this article we propose a quantum version of Shannon's conditional entropy. Given two density matrices ρ and σ on a finite dimensional Hilbert space and with S(ρ) = —Tr ρ ln ρ being the usual von Neumann entropy, this quantity S(ρ | σ) is concave in ρ and satisfies 0 ≤ S(ρ | σ) ≤ S(ρ), a quantum analogue of Shannon's famous inequality. Thus we view S(ρ | σ) as the entropy of ρ conditioned by σ. The second inequality is an equality if σ is a multiple of the identity. In contrast to the classical case, however, S(ρ | ρ) = 0 if and only if the non-vanishing eigenvalues of ρ are all non-degenerate. Also in general and again in contrast to the corresponding classical situation S(ρ, σ) = S(σ) + S(ρ | σ) is not symmetric in ρ and σ even if they commute. We also show that there is no quantum version of conditional entropy in terms of two density matrices, which shares more properties with the classical case and which in particular reduces to the classical case when the two density matrices commute. As an alternative we propose to use spectral resolutions of the unit matrix instead of density matrices. We briefly compare this with the algebraic approach of Connes and Størmer and Connes, Narnhofer and Thirring.
Abstract In this paper, we continue the investigation of quantum Markov states (QMSs) and define their mean entropies. Such entropies are explicitly computed under certain conditions. The present work takes a huge leap forward at tackling one of the most important open problems in quantum probability, which concerns the calculations of mean entropies of quantum Markov fields. Moreover, it opens up a new perspective for the generalization of many interesting results related to the one-dimensional QMSs and quantum Markov chains to multi-dimensional cases.
In this paper we compare the accessible information of quantum communication channels with the entropic content of finite-dimensional matrix algebras with respect to quantum states, as defined by Connes, Narnhofer, and Thirring. In particular, every Abelian n×n matrix algebra together with a density matrix define the input alphabet of a quantum communication channel whose accessible information equals the entropic content of the algebra with respect to the state. The cases n=2 and n=3 are concretely examined in connection with the problem of the best estimation.
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.