Stochastic Dynamics of Reduced Wave Functions and Continuous Measurement in Quantum Optics

The stochastic dynamics of open quantum systems interacting with a zero temperature environment is investigated by employing a formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. It is demonstrated that the open system dynamics can consistently be described by a stochastic process on the reduced state space. The physical meaning of reduced probability distributions on projective Hilbert space is derived from a complete, orthogonal measurement of the environment. The elimination of the variables of the environment is shown to lead to a piecewise deterministic process in Hilbert space defined by a differential Chapman-Kolmogorov equation. A Hilbert space path integral representation of the stochastic process is constructed. The general theory is illustrated by means of three examples from quantum optics. For these examples the microscopic derivation of the stochastic process is given and the general solution of the differential Chapman-Kolmogorov equation is constructed by means of the path integral representation.     

Homogeneous Decoherence Functionals¶in Standard and History Quantum Mechanics

General history quantum theories are quantum theories without a globally defined notion of time. Decoherence functionals represent the states in the history approach and are defined as certain bivariate complex-valued functionals on the space of all histories. However, in practical situations – for instance in the history formulation of standard quantum mechanics – there often is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories. In this work we study the analytic properties of (i) the standard decoherence functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restrict ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space ℋ. Among other things we prove the non-existence of a finitely valued extension for the standard decoherence functional to the space of all histories, derive a representation for the standard decoherence functional as an unbounded quadratic form with a natural representation on a Hilbert space and prove the existence of an Isham–Linden–Schreckenberg (ILS) type representation for the standard decoherence functional. Received: 26 November 1998 / Accepted: 2 December 1998     

Wasserstein distances in the analysis of time series and dynamical systems

A new approach based on Wasserstein distances, which are numerical costs of an optimal transportation problem, allows us to analyze nonlinear phenomena in a robust manner. The long-term behavior is reconstructed from time series, resulting in a probability distribution over phase space. Each pair of probability distributions is then assigned a numerical distance that quantifies the differences in their dynamical properties. From the totality of all these distances a low-dimensional representation in a Euclidean space is derived, in which the time series can be classified and statistically analyzed. This representation shows the functional relationships between the dynamical systems under study. It allows us to assess synchronization properties and also offers a new way of numerical bifurcation analysis.The statistical techniques for this distance-based analysis of dynamical systems are presented, filling a gap in the literature, and their application is discussed in a few examples of datasets arising in physiology and neuroscience, and in the well-known Hénon system.     

Wave scattering from a periodic random surface generated by a stationary binary sequence

This paper studies the scattering of a TE plane wave from a periodic random surface generated by a stochastic binary sequence using a stochastic functional method. The scattered wave is first expressed as a product of an exponential phase factor and a periodic stationary process. The periodic stationary process is then expressed by a harmonic series representation, that is a 'Fourier series' with 'Fourier coefficients' given by mutually correlated stationary processes. These stationary processes are regarded as stochastic functionals of the binary sequence and they are represented by orthogonal binary functional expansions with band-limited binary kernels. The binary kernels are determined up to the second order from the boundary condition. Then, several statistical properties of the scattering are calculated numerically and illustrated in figures. It is found that, in the binary case, the second-order scattering cross section has a subtractive term and becomes much smaller than the first-order one.     

Correct Path-Integral Formulation of Quantum Thermal Field Theory in Coherent State Representation

The path-integral quantization of thermal scalar, vector, and spinor fields is performed newly in the coherent-state representation. In doing this, we choose the thermal electrodynamics and φ⁴ theory as examples. By this quantization, correct expressions of the partition functions and the generating functionals for the quantum thermal electrodynamics and φ⁴ theory are obtained in the coherent-state representation. These expressions allow us to perform analytical calculations of the partition functions and generating functionals and therefore are useful in practical applications. Especially, the perturbative expansions of the generating functionals are derived specifically by virtue of the stationary-phase method. The generating functionals formulated in the position space are re-derived from the ones given in the coherent-state representation.     

Functional Kernel Density Estimation: Point and Fourier Approaches to Time Series Anomaly Detection

We present an unsupervised method to detect anomalous time series among a collection of time series. To do so, we extend traditional Kernel Density Estimation for estimating probability distributions in Euclidean space to Hilbert spaces. The estimated probability densities we derive can be obtained formally through treating each series as a point in a Hilbert space, placing a kernel at those points, and summing the kernels (a “point approach”), or through using Kernel Density Estimation to approximate the distributions of Fourier mode coefficients to infer a probability density (a “Fourier approach”). We refer to these approaches as Functional Kernel Density Estimation for Anomaly Detection as they both yield functionals that can score a time series for how anomalous it is. Both methods naturally handle missing data and apply to a variety of settings, performing well when compared with an outlyingness score derived from a boxplot method for functional data, with a Principal Component Analysis approach for functional data, and with the Functional Isolation Forest method. We illustrate the use of the proposed methods with aviation safety report data from the International Air Transport Association (IATA).     

部分时滞诱发Watts-Strogatz小世界神经元网络产生随机多共振

已有研究显示时滞可诱发神经元网络产生随机多共振,但它们主要讨论了神经元间的耦合都存在时滞的情形.然而实际中,有些神经元间的信息传递是瞬时的或时滞很小可以忽略的,即神经元网络中只有部分神经元间的耦合具有时滞,简称部分时滞(若神经元网络内共有l条耦合边,其中有l1条耦合边是具有时滞的,而剩余的耦合边的时滞为零,则我们称这类时滞为部分时滞).本文以Watts-Strogatz小世界神经元网络为研究对象,主要讨论部分时滞对该神经元网络系统响应强度的影响.研究结果指出,系统响应强度随部分时滞的增加呈现多峰的变化态势,即部分时滞可诱发随机多共振现象;而且使系统响应强度达到最优水平的部分时滞的取值区间随随机时滞边概率的增加渐渐变窄,当随机时滞边概率足够大时,系统响应强度只有在时滞位于外界信号周期的整数倍附近才会达到最优.此外,我们还分析了随机连边概率和神经元网络中边的总数对部分时滞诱发的随机多共振现象的影响.结果显示,部分时滞诱发的随机多共振现象对随机连边概率具有一定的鲁棒性,而神经元网络中边的总数对部分时滞诱发的随机多共振的影响则较大.     

Periodic solutions of Wick-type stochastic Korteweg–de Vries equations

Nonlinear stochastic partial differential equations have a wide range of applications in science and engineering. Finding exact solutions of the Wick-type stochastic equation will be helpful in the theories and numerical studies of such equations. In this paper, Kudrayshov method together with Hermite transform is implemented to obtain exact solutions of Wick-type stochastic Korteweg–de Vries equation. Further, graphical illustrations in two- and three-dimensional plots of the obtained solutions depending on time and space are also given with white noise functionals.     

Revisiting the European sovereign bonds with a permutation-information-theory approach

In this paper we study the evolution of the informational efficiency in its weak form for seventeen European sovereign bonds time series. We aim to assess the impact of two specific economic situations in the hypothetical random behavior of these time series: the establishment of a common currency and a wide and deep financial crisis. In order to evaluate the informational efficiency we use permutation quantifiers derived from information theory. Specifically, time series are ranked according to two metrics that measure the intrinsic structure of their correlations: permutation entropy and permutation statistical complexity. These measures provide the rectangular coordinates of the complexity-entropy causality plane; the planar location of the time series in this representation space reveals the degree of informational efficiency. According to our results, the currency union contributed to homogenize the stochastic characteristics of the time series and produced synchronization in the random behavior of them. Additionally, the 2008 financial crisis uncovered differences within the apparently homogeneous European sovereign markets and revealed country-specific characteristics that were partially hidden during the monetary union heyday.     

Direct measurement theory, Ito-Stratonovich theory, and weakly dissipative Kolmogorov-Arnold-Moser theory

The direct measurement theory studies linear functionals as applied to the problems of quantum mechanics in addition to considering quadratic functionals on the space of wave functions, well established since the beginning of the 20th century. The theory is based on the time invariance principle of an appropriate space for linear functionals. In this case, it turns out that the second-order Schr?dinger equation is factorized: factors “respect” the effect of one of two groups, i.e., the group of inertial gas motion or the nonlinear group. In the weakly dissipative Kolmogorov-Arnold-Moser (KAM) theory, the former group is of extraordinary interest in connection with the formation of caustic curves which, in turn, cause the appearance of advanced and delayed potentials, which makes it possible to estimate anew the ideas of Ito-Stratonovich in the theory of stochastic processes.     

An exact representation of the space-time characteristic functional of turbulent Navier-Stokes flows with prescribed random initial states and driving forces

By adopting a formal operator viewpoint, the space-time characteristic functional associated with Navier-Stokes turbulence is expressed in terms of a linear operator acting on the space of functionals. Obtained by a simple similarity transformation of the local translation operator generated by the nonlinear terms in the Navier-Stokes equation, this operator is unitary with respect to the formal scalar product of functionals. The equivalence of this operator representation to the functional integral representation of Rosen is shown and, for Gaussian initial velocity and external force fields, some consequences of this representation are presented.     

Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

A domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension, based on the probabilistic representation of solutions as an average of suitable multiplicative functionals. Such a direct probabilistic representation requires generating a number of random trees, whose role is that of the realizations of stochastic processes used in the linear problems. First, only few values of the sought solution inside the space-time domain are computed (by a Monte Carlo method on the trees). An interpolation is then carried out, in order to approximate interfacial values of the solution inside the domain. Thus, a fully decoupled set of sub-problems is obtained. The algorithm is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Pruning the trees is shown to increase appreciably the efficiency of the algorithm. Numerical examples conducted in 2D, including some for the KPP equation, are given.     

Data-Driven Model Reduction for Stochastic Burgers Equations

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable’s trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system’s mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model.     

Stochastic joint time–frequency response analysis of nonlinear structural systems

A novel approximate analytical approach for determining the response evolutionary power spectrum (EPS) of nonlinear/hysteretic structural systems subject to stochastic excitation is developed. Specifically, relying on the theory of locally stationary processes and utilizing a recently proposed representation of non-stationary stochastic processes via wavelets, a versatile formula for determining the nonlinear system response EPS is derived; this is done in conjunction with a stochastic averaging treatment of the problem and by resorting to the orthogonality properties of harmonic wavelets. Further, the nonlinear system non-stationary response amplitude probability density function (PDF), which is required as input for the developed approach, is determined either by utilizing a numerical path integral scheme, or by employing a time-dependent Rayleigh PDF approximation technique. A significant advantage of the approach relates to the fact that it is readily applicable for treating not only separable but non-separable in time and frequency EPS as well. The hardening Duffing and the versatile Preisach (hysteretic) oscillators are considered in the numerical examples section. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the approach.     

On Distributions of Functionals of Anomalous Diffusion Paths

Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrödinger equation in imaginary time. In recent years there is a growing interest in particular functionals of non-Brownian motion, or anomalous diffusion, but no equation existed for their PDF. Here, we derive a fractional generalization of the Feynman-Kac equation for functionals of anomalous paths based on sub-diffusive continuous-time random walk. We also derive a backward equation and a generalization to Lévy flights. Solutions are presented for a wide number of applications including the occupation time in half space and in an interval, the first passage time, the maximal displacement, and the hitting probability. We briefly discuss other fractional Schrödinger equations that recently appeared in the literature.     

A canonical dilation of the Schrödinger equation

In this paper we shall re-visit the well-known Schrödinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt. And so the unitary dynamics of a quantum object, described by the Schrödinger equation, may be obtained as the expectation of a counting process of object-clock interactions.     

A new estimator method for GARCH models

The GARCH (p, q) model is a very interesting stochastic process with widespread applications and a central role in empirical finance. The Markovian GARCH (1, 1) model has only 3 control parameters and a much discussed question is how to estimate them when a series of some financial asset is given. Besides the maximum likelihood estimator technique, there is another method which uses the variance, the kurtosis and the autocorrelation time to determine them. We propose here to use the standardized 6th moment. The set of parameters obtained in this way produces a very good probability density function and a much better time autocorrelation function. This is true for both studied indexes: NYSE Composite and FTSE 100. The probability of return to the origin is investigated at different time horizons for both Gaussian and Laplacian GARCH models. In spite of the fact that these models show almost identical performances with respect to the final probability density function and to the time autocorrelation function, their scaling properties are, however, very different. The Laplacian GARCH model gives a better scaling exponent for the NYSE time series, whereas the Gaussian dynamics fits better the FTSE scaling exponent.     

Representations of Hermitian Kernels¶by Means of Krein Spaces.¶II. Invariant Kernels

In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels that realize the given action by bounded operators on a Kre?n space. This is motivated by the GNS representation of *-algebras associated to hermitian functionals, the dilation theory of hermitian maps on C ^*-algebras, as well as others. We explain the key role played by the technique of induced Kre?n spaces and a lifting property associated to them. Received: 27 March 2000/ Accepted: 5 September 2000     

A Quantum-Bayesian Route to Quantum-State Space

In the quantum-Bayesian approach to quantum foundations, a quantum state is viewed as an expression of an agent’s personalist Bayesian degrees of belief, or probabilities, concerning the results of measurements. These probabilities obey the usual probability rules as required by Dutch-book coherence, but quantum mechanics imposes additional constraints upon them. In this paper, we explore the question of deriving the structure of quantum-state space from a set of assumptions in the spirit of quantum Bayesianism. The starting point is the representation of quantum states induced by a symmetric informationally complete measurement or SIC. In this representation, the Born rule takes the form of a particularly simple modification of the law of total probability. We show how to derive key features of quantum-state space from (i) the requirement that the Born rule arises as a simple modification of the law of total probability and (ii) a limited number of additional assumptions of a strong Bayesian flavor.     

空间激光通信中孔径平均效应的计算仿真

为了评估孔径平均效应对空间激光通信系统性能的影响,理论分析和数值计算了激光通信系统中传输链路的统计特性,包括闪烁指数、衰落概率、衰落次数和平均衰落时间。结果表明:随着接收天线孔径的增加,传输链路的闪烁指数降低、衰落概率减小、衰落次数减少、平均衰落时间缩短,孔径平均效应使得通信系统性能得到提升。由于强湍流条件下湍流尺寸小,因此孔径平均效应明显。