Quantum neural network

In the neural network theory content-addressable memories are defined by patterns that are attractors of the dynamical rule of the system. This paper develops a quantum neural network starting from a classical neural network Hamiltonian and using a Schrödinger-like equation. It then shows that such a system exhibits probabilistic memory storage characteristics analogous to those of the dynamical attractors of classical systems.     

Quantum Chaos and Dynamical Entropy

We review the notion of dynamical entropy by Connes, Narnhofer and Thirring and relate it to Quantum Chaos. A particle in a periodic potential is used as an example. This is worked out in the classical and the quantum mechanical framework, for the single particle as well as for the corresponding gas. The comparison does not only support the general assertion that quantum mechanics is qualitatively less chaotic than classical mechanics. More specifically, the same dynamical mechanism by which a periodic potential leads to a positive dynamical entropy of the classical particle may reduce the dynamical entropy of the quantum gas in comparison to free motion. Received: 26 June 1997 / Accepted: 13 April 1998     

Physical Complexity of Classical and Quantum Objects and Their Dynamical Evolution From an Information-Theoretic Viewpoint

Charles Bennett's measure of physical complexity for classical objects, namely logical-depth, is used to prove that a chaotic classical dynamical system is not physically complex. The natural measure of physical complexity for quantum objects, quantum logical-depth, is then introduced to prove that a chaotic quantum dynamical system too is not physically complex.     

Quantum chaos of the two-level atom

The quantum theory of the two-level atom coupled to a single mode of the electromagnetic field is considered as a simple example of “quantum chaos”, defined as the quantum behavior of a dynamical system which is non-integrable in the classical limit.     

二维形变谐振子系统中的量子经典对应

通过引入等效普朗克常数,将量子系统中基本动力学变量的期望值和经典系统中基本动力学变量的精确值的时间演化行为相比较,分析了两者产生差异的因素,规则运动主要是和量子效应有关,而混沌运动则是和动力学效应有关,即与系统的动力学对称性破坏相联系.在此基础上,比较了量子相空间测不准度和李雅谱诺夫指数,给出了令人满意的说明.     

Quantum statistical determinism

This paper attempts to analyze the concept of quantum statistical determinism. This is done after we have clarified the epistemic difference between causality and determinism and discussed the content of classical forms of determinism—mechanical and dynamical. Quantum statistical determinism transcends the classical forms, for it expresses the multiple potentialities of quantum systems. The whole argument is consistent with a statistical interpretation of quantum mechanics.     

Quantum Lyapunov Exponents

We show that it is possible to associate univocally with each given solution of the time-dependent Schrödinger equation a particular phase flow (quantum flow) of a non-autonomous dynamical system. This fact allows us to introduce a definition of chaos in quantum dynamics (quantum chaos), which is based on the classical theory of chaos in dynamical systems. In such a way we can introduce quantities which may be appelled quantum Lyapunov exponents. Our approach applies to a non-relativistic quantum-mechanical system of n charged particles; in the present work numerical calculations are performed only for the hydrogen atom. In the computation of the trajectories we first neglect the spin contribution to chaos, then we consider the spin effects in quantum chaos. We show how the quantum Lyapunov exponents can be evaluated and give several numerical results which describe some properties found in the present approach. Although the system is very simple and the classical counterpart is regular, the most non-stationary solutions of the corresponding Schrödinger equation are chaotic according to our definition.     

Quantum Groupoids

We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid , which gives an interpretation of the quantum dynamical Yang–Baxter equation in terms of Hopf algebroids. Received: 6 April 2000 / Accepted: 15 August 2000     

First Quantum Corrections to the Classical Partition Function for a Non-Relativistic Electrodynamical System

The classical partition function for a system in thermodynamical equilibrium formed by N identical non-relativistic particles interacting through Coulomb potentials and with the dynamical electromagnetic field is studied. It is proved that the dynamical or transverse EM degrees of freedom decouple from the particle ones. It is also shown that this decoupling does to take place in the quantum mechanical partition function. The leading quantum corrections to the classical partition function are explicitly given. Such corrections are shown to be determined by instantaneous dipole-dipole coulombic interactions and by self-energy effects, and to receive no contribution from the interaction among different particles mediated by the dynamical EM field.     

Quantum dynamical system with hyperbolic instabilities

An analysis of a quantum counterpart of a parametrically kicked nonlinear oscillator is given. The method, using as a basic criterion the recently introduced quantum characteristic exponents, is analogous to the technique developed in classical dynamical system theory. However, our approach to the characterization of the stability of an observable's evolution is done in pure quantum terms.     

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.     

Classical Limit and Quantum Logic

The analysis of the classical limit of quantum mechanics usually focuses on the state of the system. The general idea is to explain the disappearance of the interference terms of quantum states appealing to the decoherence process induced by the environment. However, in these approaches it is not explained how the structure of quantum properties becomes classical. In this paper, we consider the classical limit from a different perspective. We consider the set of properties of a quantum system and we study the quantum-to-classical transition of its logical structure. The aim is to open the door to a new study based on dynamical logics, that is, logics that change over time. In particular, we appeal to the notion of hybrid logics to describe semiclassical systems. Moreover, we consider systems with many characteristic decoherence times, whose sublattices of properties become distributive at different times.     

Witnessing Quantum Speedup through Mutual Information

Achieving dynamical speedup of evolution in an open quantum system plays a key role in many technological applications. However, how to detect quantum speedup is unclear. In this work, a method to witness quantum speedup through the measure of the mutual information is presented. It is shown that the speed of evolution of a quantum system, can be witnessed by calculating the mutual information variation, whose increase is a clear signature of dynamical speedup. The result is explained by considering the time evolution of two qubits under a one‐sided noisy channel, finding out that the mechanism for the quantum speedup is closely associated with the total exchange of information between the system and its environment, which can be expressed by the variation of mutual information. Quantitatively, the average speed of evolution is shown proportional to the average variation of the mutual information in an interval of time. The conclusion can not only explain why the quantum entanglement (or quantum coherence) and the classical correlation are neither necessary nor sufficient to speed up the quantum evolution, but also give a vital way of detecting quantum speedup in realistic environments.     

Quantum entanglement and chaos in kicked two-component Bose-Einstein condensates

We study two-component Bose-Einstein condensates that behave collectively as a spin system obeying the dynamics of a quantum kicked top. Depending on the nonlinear interaction between atoms in the classical limit, the kicked top exhibits both regular and chaotic dynamical behavior. The quantum entanglement is physically meaningful if the system is viewed as a bipartite system, where the subsystem is any one of the two modes. The dynamics of the entanglement between the two modes in this classical chaotic system has been investigated. The chaos leads to rapid rise and saturation of the quantum entanglement. Furthermore, the saturated values of the entanglement fall short of its maximum. The mean entanglement has been used to clearly display the close relation between quantum entanglement and underlying chaos.     

Higher Order Quantum Onsager Coefficients from Dynamical Invariants

Functionals representing dynamical invariants under unitary quantum dynamics of open systems are used to derive Onsager coefficients for entropy production in irreversible processes if the nonunitary time evolution is determined by quantum dynamical semigroups. The procedure allows a derivation from first principles of the quantum analogue to the classical case.     

Quantum monodromy and molecular spectroscopy

Monodromy (or once round) is a classical property of integrable dynamical systems in two or more degrees of freedom, which imposes a characteristic pattern on the quantum mechanical eigenvalue distribution. This article explains the connection by showing how the presence of an isolated critical point of the Hamiltonian leads to a classical action function that is multi-valued with respect to energy and angular momentum. Consequently, by the Bohr correspondence principle between actions and quantum numbers, there can be no uniquely defined global system of quantum numbers. Implications for the interpretation of highly excited molecular spectra are brought out by reference to quasi-linear molecules, which transfer one degree of freedom from rotational to vibrational motion during the excitation process. Emphasis is placed on the simplest examples, while a brief resumé of the wide scope of the quantum monodromy phenomenon is given in the final section.     

Transformation Properties of Dynamical Systems at the Quantum Level

Based on the generating functional of Green function for a dynamical system, the general equations of transformation properties at the quantum level are derived. In some cases they can be reduced to the quantum Noether theorem. In some other cases they can be reduced to momentum theorem or angular momentum theorem etc. at the quantum level. An example is presented and it shows that the classical conservation laws don’t always preserve in quantum theories. PACS: 11.10.E     

Quantum dynamics, measurement and entropy

The aim of this paper is to present a line of ideas, centred around entropy production andquantum dynamics, emerging from von Neumann's work on foundations of quantum mechanics and leading to current research. The concepts of measurement, dynamical evolution and entropy were central in J. von Neumann's work. Further developments led to the introduction of generalized measurements in terms of positive operator-valued measures, closely connected to the theory of open systems. Fundamental properties of quantum entropy were derived and Kolmogorov and Sinai related the chaotic properties of classical dynamical systems with asymptotic entropy production. Finally, entropy production in quantum dynamical systems was linked with repeated measurement processes and a whole research area on nonequilibrium phenomena in quantum dynamical systems seems to emerge.