We propose a mechanism by which dynamical randomness may appear in many-body systems without sensitivity to initial conditions. We characterize dynamical randomness using the Kolmogorov-Sinai (KS) entropy per unit time and its 1987 quantum extension by Connes, Narnhofer, and Thirring (CNT). In this respect, quantum systems fall into (at least) two large classes: (1) few-body bounded systems with discrete energy spectra whose entropy per unit time is vanishing; and (2) many-body open systems of statistical mechanics with continuous energy spectra whose entropy per unit time may be positive, like ideal gases and beams of particles. We calculate the classical and quantum entropies per unit time of the ideal gases and show that they are positive and in correspondence. Moreover, we show that the KS entropy per unit time is recovered from the CNT entropy per unit time using a decomposition suggested by the works of Griffiths, Gell-Mann, Hartle, and Omnès on quantum histories. We also discuss about the relationships between the quantum histories, the exponential decay of some n-time correlation functions, repeated measurements, and the CNT entropy per unit time.
We propose a mechanism by which dynamical randomness may appear in many-body systems without sensitivity to initial conditions. We characterize dynamical randomness using the Kolmogorov-Sinai (KS) entropy per unit time and its 1987 quantum extension by Connes, Narnhofer, and Thirring (CNT). In this respect, quantum systems fall into (at least) two large classes: (1) few-body bounded systems with discrete energy spectra whose entropy per unit time is vanishing; and (2) many-body open systems of statistical mechanics with continuous energy spectra whose entropy per unit time may be positive, like ideal gases and beams of particles. We calculate the classical and quantum entropies per unit time of the ideal gases and show that they are positive and in correspondence. Moreover, we show that the KS entropy per unit time is recovered from the CNT entropy per unit time using a decomposition suggested by the works of Griffiths, Gell-Mann, Hartle, and Omnès on quantum histories. We also discuss about the relationships between the quantum histories, the exponential decay of some n-time correlation functions, repeated measurements, and the CNT entropy per unit time.
We propose a mechanism by which dynamical randomness may appear in many-body systems without sensitivity to initial conditions. We characterize dynamical randomness using the Kolmogorov-Sinai (KS) entropy per unit time and its 1987 quantum extension by Connes, Narnhofer, and Thirring (CNT). In this respect, quantum systems fall into (at least) two large classes: (1) few-body bounded systems with discrete energy spectra whose entropy per unit time is vanishing; and (2) many-body open systems of statistical mechanics with continuous energy spectra whose entropy per unit time may be positive, like ideal gases and beams of particles. We calculate the classical and quantum entropies per unit time of the ideal gases and show that they are positive and in correspondence. Moreover, we show that the KS entropy per unit time is recovered from the CNT entropy per unit time using a decomposition suggested by the works of Griffiths, Gell-Mann, Hartle, and Omnès on quantum histories. We also discuss about the relationships between the quantum histories, the exponential decay of some n-time correlation functions, repeated measurements, and the CNT entropy per unit time.
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).
A new definition of the entropy of a given dynamical system and of an instrument describing the measurement process is proposed within the operational approach to quantum mechanics. It generalizes other definitions of entropy, in both the classical and quantum cases. The Kolmogorov-Sinai (KS) entropy is obtained for a classical system and the sharp measurement instrument. For a quantum system and a coherent states instrument, a new quantity, coherent states entropy, is defined. It may be used to measure chaos in quantum mechanics. The following correspondence principle is proved: the upper limit of the coherent states entropy of a quantum map as h-+0 is less than or equal to the KS-entropy of the corresponding classical map.
A new definition of the entropy of a given dynamical system and of an instrument describing the measurement process is proposed within the operational approach to quantum mechanics. It generalizes other definitions of entropy, in both the classical and quantum cases. The Kolmogorov–Sinai (KS) entropy is obtained for a classical system and the sharp measurement instrument. For a quantum system and a coherent states instrument, a new quantity, coherent states entropy, is defined. It may be used to measure chaos in quantum mechanics. The following correspondence principle is proved: the upper limit of the coherent states entropy of a quantum map as ℏ→0 is less than or equal to the KS-entropy of the corresponding classical map. ‘‘Chaos umpire sits, And by decision more imbroils the fray By which he reigns: next him high arbiter Chance governs all.’’ John Milton, Paradise Lost, Book II
I give an informal overview of the decoherent histories approach to quantum mechanics, due to Griffiths, to Omnès, and to Gell-Mann and Hartle is given. Results on the connections between decoherence, records, correlation and entropy are described. The emphasis of the presentation is on understanding the broader meaning of the conditions of consistency and decoherence, and in particular, the extent to which they permit one to assign definite properties to the system. The quantum Brownian motion model is briefly discussed. (To appear in proceedings of the workshop, "Stochastic Evolution of Quantum States in Open Systems and Measurement Processes", Budapest, March, 1993, edited by L.Diosi).
Remarks concerning an Entropy-like Quantity Sq especially for Quantum Systems with a Particle Picture; Model with pure Point Spectra
For quantum mechanical systems an entropy-like quantity Sq is defined. Sq can differ from the usually defined entropy S and Sq may increase with time for an isolated system. The essential condition for the difference between S and Sq is the assumption that the set {\bf A} of observables which can be represented by a measurement is a proper subset of the set of selfadjoint operators. The underlying idea is made visible in the case of particle systems with non-trivial scattering. The model-character of the reasoning comes from the fact that continuous spectra are replaced by point-spectra. So it seems evident, that no direct connection exists between Sq and the Sinai-Kolmogorov-Entropy at least in this model with pure point-spectra.