Classical Random Field Research Articles

Characterization of Entanglement via Non‐Existence of a Subquantum Random Field

AbstractAny pure state of a compound system with the state space determines a kind of covariance operator acting in the Cartesian product . If this operator is positively defined, then it determines a random field valued in . In this case compound quantum system can be treated as a classical random field system whose configuration space is not tensor, but Cartesian product space. It happens that and a subquantum process exists if and only if quantum state is not entangled. The technical framework used in this note is already presented by von Neumann.

Regularized Loss With Hyperparameter Estimation for Weakly Supervised Single Class Segmentation.

We propose a new image level weakly supervised segmentation approach for datasets with a single object class of interest. Our approach is based on a regularized loss function inspired by the classical Conditional Random Field (CRF) modeling. Our loss models properties of generic objects, and we use it to guide CNN towards segments that are more likely to correspond to the object, thus avoiding the need for pixel precise annotations. Training CNN with regularized loss is a difficult task for gradient descent. We develop an annealing algorithm which is crucial for a successful training. Furthermore, we develop an approach for hyperparameter setting for the most important components of our regularized loss. This is far from trivial, since there is no pixel precise ground truth for guidance. The advantage of our method is that we use a standard CNN architecture and an easy to interpret loss function, derived from classical CRF models. Furthermore, we apply the same loss function for any task/dataset. We first evaluate our approach for salient object segmentation and co-segmentation. These tasks naturally involve one object class of interest. Then we adapt our approach to image level weakly supervised multi-class semantic segmentation. We obtain state-of-the-art results.

Evaluating statistical homogeneity of cone penetration test (CPT) data profile using auto-correlation function

Cone penetration test (CPT) data profile is often numerically modelled using random field theory and based implicitly on a fundamental assumption: statistical homogeneity of the preprocessed stationary CPT data profile. In classical random field theory, statistical homogeneity evaluation may only be achieved if the CPT data profile is infinite in length. While in engineering practice, it is often the case that a CPT data profile has limited sounding depth. This leads to a critical question that whether the CPT data profile satisfies statistical homogeneity assumption to enable proper random field modelling. This paper proposes a novel non-parametric method for evaluating statistical homogeneity of a single CPT data profile. A novel unified representation of CPT data profile pattern is developed in this study using discrete cosine transform (DCT)-based auto-correlation function (ACF). Statistical homogeneity evaluation of a CPT data profile is reformulated as statistical analysis of pattern similarity between the original CPT data profile and its partial segment profiles. The proposed method is general since it is distribution-free and does not require a parametric correlation function. The proposed method performs well on both simulated and real examples and can identify the possible locations of heterogeneity in a CPT profile.

Machine learning based priors for Bayesian inversion in MR imaging

The Bayesian approach allows the incorporation of informative prior knowledge to effectively enable and improve the solution of inverse problems. Obtaining prior information in probabilistic terms is, however, a challenging task. Recently, machine learning has been applied for the training of generative models to facilitate the translation of historically or otherwise available data to a prior distribution. In this work, we apply this methodology to undersampled magnetic resonance imaging. In particular, we employ an autoencoder as part of a generative model to statistically regularise and solve the high-dimensional inverse problem using Bayesian inversion. Comparison with a classical Gaussian Markov random field prior is performed and numerical examples highlight the possible advantages of data-driven priors.

Random Ising chain in transverse and longitudinal fields: Strong disorder RG study

Motivated by the compound LiHoxY1-xF4, we consider the Ising chain with random couplings and in the presence of simultaneous random transverse and longitudinal fields, and study its low-energy properties at zero temperature by the strong disorder renormalization group approach. In the absence of longitudinal fields, the system exhibits a quantum-ordered and a quantum-disordered phase separated by a critical point of infinite disorder. When the longitudinal random field is switched on, the ordered phase vanishes and the trajectories of the renormalization group are attracted to two disordered fixed points: one is characteristic of the classical random field Ising chain, the other describes the quantum disordered phase. The two disordered phases are separated by a separatrix that starts at the infinite disorder fixed point and near which there are strong quantum fluctuations.

Transformed Gaussian Random Fields for Unsupervised Image Deconvolution

This paper deals with the problem of Bayesian deconvolution. Starting from the classical Gaussian Markov Random Fields (GMRF) prior, we present a broader model referred as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">transformed</i> GMRF (TGRF) in which the latent field results from a nonlinear transformation of a GMRF. We propose a Bayesian inference method to estimate TGRF from an observed image with known parameters, and introduce methods inspired from expectation-maximization in order to jointly deconvolve and estimate the statistical model's parameter for both GMRF and TGRF. Numerical results allow to determine the best inference method among several possibilities on fully synthetic data, on a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">phantom</i> image and on real fluorescence microscopy images.

Dynamic mean-field theory for dense spin systems at infinite temperature

A dynamic mean-field theory for spin ensembles (spinDMFT) at infinite temperatures on arbitrary lattices is established. The approach is introduced for an isotropic Heisenberg model with $S=\frac{1}{2}$ and external field. For large coordination numbers, it is shown that the effect of the environment of each spin is captured by a classical time-dependent random mean field which is normally distributed. Expectation values are calculated by averaging over these mean fields, i.e., by a path integral over the normal distributions. A self-consistency condition is derived by linking the moments defining the normal distributions to spin autocorrelations. In this framework, we explicitly show how the rotating-wave approximation becomes a valid description for increasing magnetic field. We also demonstrate that the approach can easily be extended. Exemplarily, we employ it to reach a quantitative understanding of a dense ensemble of spins with dipolar interaction which are distributed randomly on a plane including static Gaussian noise as well.

Effects of classical random external field on the dynamics of entanglement in a four-qubit system

We investigate the dynamics of entanglement through negativity and witness operators in a system of four non-interacting qubits driven by a classical phase noisy laser characterized by a classical random external field (CREF). The qubits are initially prepared in the GHZ-type and W-type states and interact with the CREF in two different qubit-field configurations, namely, common environment and independent environments in which the cases of equal and different field phase probabilities are distinguished. We find that entanglement exhibits different decaying behavior, depending on the input states of the qubits, the qubit-field coupling configuration, and field phase probabilities. On the one hand, we demonstrate that the coupling of the qubits in a common environment is an alternative and more efficient strategy to completely shield the system from the detrimental impacts of the decoherence process induced by a CREF, independent of the input state and the field phase probabilities considered. Also, we show that GHZ-type states have strong dynamics under CREF as compared to W-type states. On the other hand, we demonstrate that in the model investigated the system robustness’s can be greatly improved by increasing the number of qubits constituting the system.

Dynamics and protection of quantum correlations in a qubit–qutrit system subjected locally to a classical random field and colored noise

This paper investigates the evolution and protection of quantum correlations (entanglement and quantum discord (QD) in particular), in a hybrid system formed by a qubit subjected to an external random field and a qutrit subjected to a colored noise generated either by a single or collection of many random bistable fluctuators (RBFs). Two different input states namely the one- and two-parameter class of states of qubit–qutrit are investigated. Entanglement and QD are quantified by means of negativity and geometric QD and their protection investigated by recourse to the weak measurement (WM) and weak measurement reversal (WMR) technique. It is shown that the immunity of entanglement and QD against decoherence can be efficiently increased by properly adjusting the parameters of the input states, no matter the spectrum of the colored noise and the number of RBFs considered. Moreover, it is shown that the WM and WMR technique fails in protecting entanglement in the studied system, but can effectively protect QD from the detrimental impacts of the decoherence. In fact, it is shown that performing a WM followed by a WMR on the qubit of the system allows to shield the QD of the system from decoherence even when the decoherence process is strengthen by considering a large number of RBFs affecting the qutrit.

Certified and fast computations with shallow covariance kernels

Many techniques for data science and uncertainty quantification demand efficient tools to handle Gaussian random fields, which are defined in terms of their mean functions and covariance operators. Recently, parameterized Gaussian random fields have gained increased attention, due to their higher degree of flexibility. However, especially if the random field is parameterized through its covariance operator, classical random field discretization techniques fail or become inefficient. In this work we introduce and analyze a new and certified algorithm for the low-rank approximation of a parameterized family of covariance operators which represents an extension of the adaptive cross approximation method for symmetric positive definite matrices. The algorithm relies on an affine linear expansion of the covariance operator with respect to the parameters, which needs to be computed in a preprocessing step using, e.g., the empirical interpolation method. We discuss and test our new approach for isotropic covariance kernels, such as Mat\'ern kernels. The numerical results demonstrate the advantages of our approach in terms of computational time and confirm that the proposed algorithm provides the basis of a fast sampling procedure for parameter dependent Gaussian random fields.

Classical states, quantum field measurement

Classical Koopman–von Neumann Hilbert spaces of states are constructed here by the action of classical random fields on a vacuum state in ways that support an action of the quantized electromagnetic field and of the U(1)—invariant observables of the quantized Dirac spinor field, allowing a manifestly Lorentz invariant classical understanding of the state spaces of the two field theories, generalizing the Quantum–Mechanics–Free Systems of Tsang&Caves and Quantum Non-Demolition measurements. The algebra of functions on a classical phase space is commutative but the algebra of classical observables associated with coordinate transformations is noncommutative, so that, for example, we can as much ask whether a classical state is an eigenstate of a rotation as we can in quantum mechanics and so that entangled states can be distinguished from mixed states, making classical random fields as weird as quantum fields.

Faster family-wise error control for neuroimaging with a parametric bootstrap.

In neuroimaging, hundreds to hundreds of thousands of tests are performed across a set of brain regions or all locations in an image. Recent studies have shown that the most common family-wise error (FWE) controlling procedures in imaging, which rely on classical mathematical inequalities or Gaussian random field theory, yield FWE rates (FWER) that are far from the nominal level. Depending on the approach used, the FWER can be exceedingly small or grossly inflated. Given the widespread use of neuroimaging as a tool for understanding neurological and psychiatric disorders, it is imperative that reliable multiple testing procedures are available. To our knowledge, only permutation joint testing procedures have been shown to reliably control the FWER at the nominal level. However, these procedures are computationally intensive due to the increasingly available large sample sizes and dimensionality of the images, and analyses can take days to complete. Here, we develop a parametric bootstrap joint testing procedure. The parametric bootstrap procedure works directly with the test statistics, which leads to much faster estimation of adjusted p-values than resampling-based procedures while reliably controlling the FWER in sample sizes available in many neuroimaging studies. We demonstrate that the procedure controls the FWER in finite samples using simulations, and present region- and voxel-wise analyses to test for sex differences in developmental trajectories of cerebral blood flow.

Scaling analysis and instantons for thermally assisted tunneling and quantum Monte Carlo simulations

We develop an instantonic calculus to derive an analytical expression for the thermally-assisted tunneling decay rate of a metastable state in a fully connected quantum spin model. The tunneling decay problem can be mapped onto the Kramers escape problem of a classical random dynamical field. This dynamical field is simulated efficiently by path integral Quantum Monte Carlo (QMC). We show analytically that the exponential scaling with the number of spins of the thermally-assisted quantum tunneling rate and the escape rate of the QMC process are identical. We relate this effect to the existence of a dominant instantonic tunneling path. The instanton trajectory is described by nonlinear dynamical mean-field theory equations for a single site magnetization vector, which we solve exactly. Finally, we derive scaling relations for the "spiky" barrier shape when the spin tunnelling and QMC rates scale polynomially with the number of spins $N$ while a purely classical over-the-barrier activation rate scales exponentially with $N$.

Compositional phase diagram and microscopic mechanism of Ba1-xCaxZryTi1-yO3 relaxor ferroelectrics.

With extensive first-principles density-functional calculations, we construct a three-dimensional compositional phase diagram of Ba1-xCaxZryTi1-yO3 (BCZT) with the Ca and Zr content in the ranges of 0 ≤ xCa ≤ 0.2 and 0 ≤ yZr ≤ 1. Our calculations show that, when the Zr content increases, the difference in energy and difference in the structural parameters of the cubic, tetragonal, orthorhombic, and rhombohedral phases of BCZT are reduced. Eventually, all four phases merge into a multiphase with coexisting cubic structures (MPCCS) under Zr-rich conditions, indicating that BCZT undergoes phase transition from a normal ferroelectric (NFE) to a relaxor ferroelectric (RFE), consistent with experimental observations. The 3D diagram shows that the regions of merged and separated energy surfaces correspond to the regions of the RFE and NFE, respectively, which suggests that a MPCCS corresponds to a RFE. In addition, with the MPCCS model and Landau-Devonshire theory, we provide an interpretation of the high electromechanical properties of the BCZT relaxor ferroelectric and apply it to the classical local random field and micro-macro domain transition models.

Emergence of Quantum Mechanics from Theory of Random Fields

Our aim in this paper is to enlighten the possibility to treat quantum mechanics as emergent from a kind of classical physical model, in spite of recent remarkable experiments demonstrating a violation of the Bell inequality. To proceed in a rigorous way, we use the methodology of ontic–epistemic modeling of physical phenomena. This methodology is rooted in the old Bild conception about theoretical and observational models in physics. This conception was elaborated in the fundamental works of Hertz, Boltzmann, and Schrodinger. Our ontic model (generating the quantum model) is of the random field type, prequantum classical statistical field theory (PCSFT). We present a brief review of its basic features without overloading the presentation by mathematical details. Then we show that the Bell inequality can be violated not only at the epistemic level, i.e., for observed correlations, but even at the ontic level, for classical random fields. We devote the important part of the paper to an analysis of the internal energy structure of prequantum random fields and their coupling with the background field of subquantum fluctuations. Finally, we present a unified picture of the microworld based on the composition of prequantum random fields from elementary fluctuations. Since quantum systems are treated as the symbolic representation of prequantum fields, this picture leads to a unifying treatment of all quantum systems as special blocks of elementary fluctuations carrying negligibly small energies.

Quantum epistemology from subquantum ontology: Quantum mechanics from theory of classical random fields

The scientific methodology based on two descriptive levels, ontic (reality as it is ) and epistemic (observational), is briefly presented. Following Schrödinger, we point to the possible gap between these two descriptions. Our main aim is to show that, although ontic entities may be unaccessible for observations, they can be useful for clarification of the physical nature of operational epistemic entities. We illustrate this thesis by the concrete example: starting with the concrete ontic model preceding quantum mechanics (the latter is treated as an epistemic model), namely, prequantum classical statistical field theory (PCSFT), we propose the natural physical interpretation for the basic quantum mechanical entity—the quantum state (“wave function”). The correspondence PCSFT ↦ QM is not straightforward, it couples the covariance operators of classical (prequantum) random fields with the quantum density operators. We use this correspondence to clarify the physical meaning of the pure quantum state and the superposition principle—by using the formalism of classical field correlations.

Stochastic quantum Zeno-based detection of noise correlations.

A system under constant observation is practically freezed to the measurement subspace. If the system driving is a random classical field, the survival probability of the system in the subspace becomes a random variable described by the Stochastic Quantum Zeno Dynamics (SQZD) formalism. Here, we study the time and ensemble average of this random survival probability and demonstrate how time correlations in the noisy environment determine whether the two averages do coincide or not. These environment time correlations can potentially generate non-Markovian dynamics of the quantum system depending on the structure and energy scale of the system Hamiltonian. We thus propose a way to detect time correlations of the environment by coupling a quantum probe system to it and observing the survival probability of the quantum probe in a measurement subspace. This will further contribute to the development of new schemes for quantum sensing technologies, where nanodevices may be exploited to image external structures or biological molecules via the surface field they generate.

Random fields of bounded variation and computation of their variation intensity

AbstractThe main purpose of this paper is to define and characterize random fields of bounded variation, that is, random fields with sample paths in the space of functions of bounded variation, and to study their mean total variation. Simple formulas are obtained for the mean total directional variation of random fields, based on known formulas for the directional variation of deterministic functions. It is also shown that the mean variation of random fields with stationary increments is proportional to the Lebesgue measure, and an expression of the constant of proportionality, called thevariation intensity, is established. This expression shows, in particular, that the variation intensity depends only on the family of two-dimensional distributions of the stationary increment random field. When restricting to random sets, the obtained results give generalizations of well-known formulas from stochastic geometry and mathematical morphology. The interest of these general results is illustrated by computing the variation intensities of several classical stationary random field and random set models, namely Gaussian random fields and excursion sets, Poisson shot noises, Boolean models, dead leaves models, and random tessellations.

Sudden death and rebirth of entanglement for different dimensional systems driven by a classical random external field

The entangled behavior of different dimensional systems driven by a classical external random field (CERF) is investigated. The amount of the survival entanglement between the components of each system is quantified. There are different behaviors of entanglement that come into view: decay, sudden death, sudden birth and long-lived entanglement. The maximum entangled states which can be generated from any of theses suggested systems are much more fragile than the partially entangled ones. The systems of larger dimensions are more robust than those of smaller dimension systems, where the entanglement decays smoothly and gradually and may vanish for a very short time. For the class of dimensional system, the one-parameter family is found to be more robust than the two-parameter family. Although the entanglement of a driven dimensional system is very sensitive to the CERFs, one can use them to generate a long-lived entanglement.

Distributed correlations and information flows within a hybrid multipartite quantum-classical system

Understanding the non-Markovian mechanisms underlying the revivals of quantum entanglement in the presence of classical environments is central in the theory of quantum information. Tentative interpretations have been given by either the role of the environment as a control device or the concept of hidden entanglement. We address this issue from an information-theoretic point of view. To this aim, we consider a paradigmatic tripartite system, already realized in the laboratory, made of two independent qubits and a random classical field locally interacting with one qubit alone. We study the dynamical relationship between the two-qubit entanglement and the genuine tripartite correlations of the overall system, finding that collapse and revivals of entanglement correspond, respectively, to raise and fall of the overall tripartite correlations. Interestingly, entanglement dark periods can enable plateaux of nonzero tripartite correlations. We then explain this behavior in terms of information flows among the different parties of the system. Besides showcasing the phenomenon of the freezing of overall correlations, our results provide new insights on the origin of retrieval of entanglement within a hybrid quantum-classical system.