Nonlocal Form Research Articles

Non-local Markovian Symmetric Forms on Infinite Dimensional Spaces

The general framework on the non-local Markovian symmetric forms on weighted lp(p in [1, infty ]) spaces constructed by Albeverio et al. (Commn. Math. Phys. 388, 659–706, 2021 Kagawa) by restricting the situation where p = 2, is applied to probability measure spaces describing the space cut-off P(ϕ)2 Euclidean quantum field, the 2-dimensional Euclidean quantum fields with exponential and trigonometric potentials, and the measure associated with the field describing a system of an infinite number of classical particles. For each measure space, the Markov process corresponding to the non-local type stochastic quantization is constructed.

Experimental Demonstration of Quantum Pseudotelepathy.

Quantum pseudotelepathy is a strong form of nonlocality. Different from the conventional nonlocal games where quantum strategies win statistically, e.g., the Clauser-Horne-Shimony-Holt game, quantum pseudotelepathy in principle allows quantum players to with probability 1. In this Letter, we report a faithful experimental demonstration of quantum pseudotelepathy via playing the nonlocal version of Mermin-Peres magic square game, where Alice and Bob cooperatively fill in a 3×3 magic square. We adopt the hyperentanglement scheme and prepare photon pairs entangled in both the polarization and the orbital angular momentum degrees of freedom, such that the experiment is carried out in a resource-efficient manner. Under the locality and fair-sampling assumption, our results show that quantum players can simultaneously win all the queries over any classical strategy.

Nonlocality of quantum matter corrections and cosmological constant running

Semiclassical loop corrections to the gravitational action include various terms with zero, two, and four derivatives of the metric as well as nonlocal form factors for these terms. Contributions to some of these terms could be confused with others on a specific metric background or for a particular gauge fixing. We present a critical analysis of the recent works where the tensor structure and the number of derivatives in the action of gravity were not properly taken into account. Taking these relevant aspects into account, we show that although some contributions owing to the quantum fluctuations of massive or massless scalar as well as fermion and vector fields may be attributed to the “running” of the cosmological constant, in reality they correspond to the fourth derivative terms of the action.

Solitons and symmetry reduction solutions of a nonlocal two‐mode Korteweg–de Vries equation

The two‐mode Korteweg–de Vries (TMKdV) equation describes propagation of two different waves modes simultaneously. By using consistent correlated bang (CCB) method, a nonlocal form of the TMKdV equation is constructed and converted into two local equations with its dependent variables having definite parity. Multiple soliton solutions of the nonlocal TMKdV equation are obtained by using known multiple soliton solutions of the TMKdV equation, and some singular travelling wave solutions are also generated in a similar way. The classical Lie symmetry method is also carried on the nonlocal TMKdV equation to get its symmetry reduction solutions.

Analysis of Nonlinear Heat Conduction Problems with Temperature-Dependent Conductivity Using Peridynamic Differential Operator

In this paper, a nonlocal approach for handling nonlinear heat conduction problems is developed using the peridynamic differential operator (PDDO). The boundary conditions and heat conduction equations are transformed from local differential form to nonlocal integral form by introducing peridynamic functions. The algebraic equations are established by the Lagrange multiplier method and variational analysis, which is solved with Newton–Raphson iterative method. Nonlinearities resulting from the temperature-dependence of material properties have been taken into account in irregular boundaries and cracked plates. Numerical examples demonstrate the effectiveness and accuracy of the PDDO for solving nonlinear heat conduction problems with temperature-dependent conductivity.

Quantum violation of local causality in an urban network using hybrid photonic technologies

Quantum networks play a crucial role in distributed quantum information processing, enabling the establishment of entanglement and quantum communication among distant nodes. Fundamentally, networks with independent sources allow for new forms of nonlocality, beyond the paradigmatic Bell’s theorem. Here we implement the simplest of such networks—the bilocality scenario—in an urban network connecting different buildings with a fully scalable and hybrid approach. Two independent sources using different technologies—a quantum dot and a nonlinear crystal—are used to share a photonic entangled state among three nodes connected through a 270 m free-space channel and fiber links. By violating a suitable nonlinear Bell inequality, we demonstrate the nonlocal behavior of the correlations among the nodes of the network. Our results pave the way towards the realization of more complex networks and the implementation of quantum communication protocols in an urban environment, leveraging the capabilities of hybrid photonic technologies.

Fundamentals of acoustic Willis media

The cross-coupling between strain and velocity in acoustic materials, known as Willis coupling, has recently been receiving increasing attention within the broad acoustics community. Willis coupling can provide a new degree of freedom to control sound propagation, which has been enabling several novel applications. In this work, based on the general constitutive relations of Willis media and the acoustic Poynting theorem, we study the constraints stemming from fundamental symmetries – parity, time-reversal, reciprocity and energy conservation – in these media. The wave features, such as wave-vectors, wave impedance and orthogonality are also generally investigated. In addition, we put forward a nonlocal model that unveils the relation between Willis media and nonlocal materials, and how Willis phenomena stem from weak forms of nonlocality.

Steady-State Thermodynamics of a Cascaded Collision Model.

We study the steady-state thermodynamics of a cascaded collision model where two subsystems and collide successively with an environment R in the cascaded fashion. We first formulate general expressions of thermodynamics quantities and identify the nonlocal forms of work and heat that result from cascaded interactions of the system with the common environment. Focusing on a concrete system of two qubits, we then show that, to be able to unidirectionally influence the thermodynamics of , the former interaction of should not be energy conserving. We finally demonstrate that the steady-state coherence generated in the cascaded model is a kind of useful resource in extracting work, quantified by ergotropy, from the system. Our results provide a comprehensive understanding on the thermodynamics of the cascaded model and a possible way to achieve the unidirectional control on the thermodynamics process in the steady-state regime.

General Non-Local Continuum Mechanics: Derivation of Balance Equations

In this paper, mechanics of continuum with general form of nonlocality in space and time is considered. Some basic concepts of nonlocal continuum mechanics are discussed. General fractional calculus (GFC) and general fractional vector calculus (GFVC) are used as mathematical tools for constructing mechanics of media with general form of nonlocality in space and time. Balance equations for mass, momentum, and energy, which describe conservation laws for nonlocal continuum, are derived by using the fundamental theorems of the GFC. The general balance equation in the integral form are derived by using the second fundamental theorems of the GFC. The first fundamental theorems of GFC and the proposed fractional analogue of the Titchmarsh theorem are used to derive the differential form of general balance equations from the integral form of balance equations. Using the general fractional vector calculus, the equations of conservation of mass, momentum, and energy are also suggested for a wide class of regions and surfaces.

Characterizing nonlocal correlations through various n -locality inequalities in a quantum network

The multipartite quantum networks feature multiple independent sources, in contrast to the conventional multipartite Bell experiment involving a single source. Despite the initial independence of resources, the multiple observers in the network can suitably choose measurements on their local subsystems and generate a form of quantum nonlocality across the network. So far, network nonlocality has been explored when each source produces a two-qubit entangled state. In this work we demonstrate the network nonlocality when each party performs a black-box measurement, and the dimension of the system remains unspecified. In an interesting work, by considering each source produces two-qubit entangled states in the conventional bilocal scenario, Gisin et al. [Phys. Rev. A 96, 020304 (2017)] demonstrated a correspondence between the violations of bipartite Clauser-Horne-Shimony-Halt inequality and the bilocality inequality. We introduce a variant of the sum-of-squares approach to reproduce their results without assuming the dimension of the system. We then generalize the argument for network nonlocality in star-network topology. Furthermore, we propose a set of $n$-locality inequalities in a star-network configuration where each of the $n$ parties performs an arbitrary number of dichotomic measurements and demonstrates the above correspondence between the quantum violations of the $n$-locality inequalities and the chained Bell inequalities. A similar correspondence is demonstrated based on a recently formulated family of $n$-locality inequalities whose optimal quantum violation cannot be obtained when each source emits a two-qubit entangled state and requires multiple copies of two-qubit entangled states. Throughout this paper, each party in the network performs black-box measurements, and the dimension of the system remains unspecified.

Nonlocal quadratic forms with visibility constraint

Given a subset D of the Euclidean space, we study nonlocal quadratic forms that take into account tuples \((x,y) \in D \times D\) if and only if the line segment between x and y is contained in D. We discuss regularity of the corresponding Dirichlet form leading to the existence of a jump process with visibility constraint. Our main aim is to investigate corresponding Poincaré inequalities and their scaling properties. For dumbbell shaped domains we show that the forms satisfy a Poincaré inequality with diffusive scaling. This relates to the rate of convergence of eigenvalues in singularly perturbed domains.

Infinite-derivative linearized gravity in convolutional form

This article aims to transform the infinite-order Lagrangian density for ghost-free infinite-derivative linearized gravity into non-local form. To achieve it, we use the theory of generalized functions and the Fourier transform in the space of tempered distributions . We show that the non-local operator domain is not defined on the whole functional space but on a subset of it. Moreover, we prove that these functions and their derivatives are bounded in all and, consequently, the Riemann tensor is regular and the scalar curvature invariants do not present any spacetime singularity. Finally, we explore what conditions we need to satisfy so that the solutions of the linearized equations of motion exist in .

On the domains of Dirichlet forms on metric measure spaces

We prove that the domain of the local Dirichlet form is strictly contained in the domain of any stable-like non-local Dirichlet form on general metric measure spaces. We construct explicit functions in the difference of the domains of the Dirichlet forms on the Sierpiński gasket and the Sierpiński carpet.

Network nonlocality via rigidity of token counting and color matching

Network Nonlocality is the study of the Network Nonlocal correlations created by several independent entangled states shared in a network. In this paper, we provide the first two generic strategies to produce nonlocal correlations in large classes of networks without input. In the first one, called Token-Counting (TC), each source distributes a fixed number of tokens and each party counts the number of received tokens. In the second one, called Color-Matching (CM), each source takes a color and a party checks if the color of neighboring sources match. Using graph theoretic tools and Finner's inequality, we show that TC and CM distributions are rigid in wide classes of networks, meaning that there is essentially a unique classical strategy to simulate such correlations. Using this rigidity property, we show that certain quantum TC and CM strategies produce correlations that cannot be produced classicality. This leads us to several examples of Network Nonlocality without input. These examples involve creation of coherence throughout the whole network, which we claim to be a fingerprint of genuine forms of Network Nonlocality. This work extends a more compact parallel work [Nonlocality for Generic Networks, arXiv:2011.02769] on the same subject and provides all the required technical proofs.

Solving the eikonal equation for compressional and shear waves in anisotropic media using peridynamic differential operator

SUMMARY Traveltimes of compressional (P) and shear (S) waves have proven essential in many earthquake and exploration seismology applications. An accurate and efficient traveltime computation for P and S waves is crucial for the success of these applications. However, solving the eikonal equation with a complex phase velocity field in anisotropic media is challenging. The eikonal equation is a first-order nonlinear hyperbolic partial differential equation. It represents the high-frequency asymptotic approximation of the wave equation. The fast marching and sweeping methods are commonly used due to their efficiency in numerically solving the eikonal equation. However, these methods suffer from numerical inaccuracy in anisotropic media with sharp heterogeneity, irregular surface topography and complex phase velocity fields. This study presents a new method for the solution of the eikonal equation by employing the peridynamic differential operator (PDDO). The PDDO provides the non-local form of the eikonal equation by introducing an internal length parameter (horizon) and a weight function with directional non-locality. The operator is immune to discontinuities in the form of sharp changes in field or model variables and invokes the direction of traveltime in a consistent manner. The weight function controls the strength of association among points within the horizon. Solutions are constructed in a consistent manner without upwind assumptions through simple discretization. The robustness of this approach is established by considering different types of eikonal equations on complex velocity models in anisotropic media. The examples demonstrate its unconditional numerical stability and results compare well with the reference solutions.

On the solution of hyperbolic equations using the peridynamic differential operator

Numerical solution of hyperbolic differential equations, such as the advection equation, poses challenges. Classically, this issue has been addressed by using a scheme known as the upwind scheme. It simply invokes more points from the upwind side of the flow stream when calculating derivatives. This study presents a generalized upwind scheme, referred to as directional nonlocality, for the numerical solution of linear and nonlinear hyperbolic Partial Differential Equations (PDEs) using the peridynamic differential operator (PDDO). The PDDO provides the nonlocal form of the differential equations by introducing an internal length parameter (horizon) and a weight function. The weight function controls the degree of interaction among the points within the horizon. A modification to the weight function, i.e., upwinded-weight function, accounts for directional nonlocality along which information travels. This modification results in a stable PDDO discretization of hyperbolic PDEs. Solutions are constructed in a consistent manner without special treatments through simple discretization. The capability of this approach is demonstrated by considering time dependent linear and nonlinear hyperbolic equations as well as the time invariant Eikonal equation. Numerical stability is ensured for the linear advection equation and the PD solutions compare well with the analytical/reference solutions.

Topological description of complex patterns of bonding, charge transference and structural changes in chemical reactions: SN2 type reactions, a case study

In this report we deeply explore the electron rearrangements for a chemical reaction case study of the SN2 type OH- + CH3-CN ↔ HO-CH3 + CN-, within the topological formalism of QTAIM model in its local and the non-local or integrated form implementations. This is part of our main interest of a subtle description of the electron distribution at both, equilibrium and non-equilibrium conformations for the state function at correlated level of approximation. The emphasis is mainly placed on the determination of complex patterns of interaction of two or four electrons in three centers (2e-3c, 4e-3c). These results confirm them as a measure of electron excess or electron deficiency effects and revels their crucial importance in the charge transference process and its relationship with these complex patterns. All electronic magnitudes were followed along the configurations from reactants to products passing through the corresponding transition state TS. The results obtained throughout this case study reveals the high performance of the topological formalism of Bader type in the subtle description of electron rearrangements in a chemical transformation.

Phase Field vs Gradient Enhanced Damage Models: A Comparative Study

A comparison of the two different approaches for modelling damage in material in an infinitesimal strain setting is studied. The first approach is the nonlocal gradient enhanced damage model where the damage variable is taken as an independent variable which will be determined based on the local strain measure. Here, the nonlocal integral form is approximated to an implicit or explicit differential form using the Taylor’s series expansion for simpler numerical implementation. The second approach is the phase field damage model where a Helmholtz free energy density function is considered that includes a new energy degradation function along with a phase field non-conserved order parameter. The first variational principle on this energy density functional with respect to the corresponding order parameter variable will reach a stationarity value resulting in the non-conserved Allen-Cahn equation. The relationship of the order parameter with the damage variable gives the Allen-Cahn evolution equation for damage. A 1D bar example is considered for commenting on the similarities and differences of the two approaches to damage based on the mesh convergence studies based on the results obtained for various meshing profiles, changing length scale parameter values and the obtained damage profiles.

Self-Testing of Quantum States Using Symmetric Local Hidden State Model

We introduce a symmetric local hidden state $(slhs)$ model in a scenario, where two spacially separated parties receive quantum states from an unknown source. We derive an inequality based on the model. A completely new form of nonlocality emerges from the resource theoretic point of view. The inequality singles out a larger set of quantum correlated states in the higher dimensional scenarios $(d> 2 $ X $2)$ than what is predicted by the existing $lhs$ model, opening a new front for the experimentalists to test the accuracy of the prediction. We propose an experiment to show the experimental violation of the inequality in the two qubit scenario and perform the experiment on the IBM quantum computer. However, the experimental method adopted for the two-qubit scenario does not naturally generalize in the higher dimensional scenarios and leaves the experimental verification of the claim open. We also show that the maximal violation of the inequality can be used to self-test the Bell state and measurement bases, leading to complete device-independence.

Similarity Weight Learning: A New Spatial and Temporal Satellite Image Fusion Framework

Spatiotemporal fusion is a topical framework for solving the mutual restricted problem between the spatial and temporal resolution of satellite images. We pioneer an approach to replace similarity measurement steps in spatiotemporal fusion algorithms with convolutional neural networks (CNNs), building a bridge between weight function-based models and the learning-based models. Specifically, we propose a nonlocal form that separates the relational computation part from the value representation part, and construct the CNN-based similarity weight learning block for learning normalized weights. The block can be inserted into spatial and temporal adaptive reflectance fusion model (STARFM) to replace the manually designed weight calculation rules common in weight function-based methods, or into the CNN model StfNet to better utilize neighboring high-resolution images. The trained model outputs a high-resolution prediction from each base date image pair. The final result is a combination of the two predictions. In this regard, we propose the standard deviation-based weights to combine two prediction results. Four experiments are performed on Landsat–Moderate-resolution Imaging Spectroradiometer (MODIS) image pairs to determine the following: 1) the performance of the model at the target training date; 2) the generalization of the model in the target training time period; and 3) the generalization of the model at different dates and different geographical locations, each considering the different cases of giving one and two pairs of known images. Experimental results demonstrate the superiority of the similarity weight learning block and standard deviation-based weights. Among them, STARFM with the similarity weight learning block exhibits strong generalization, which testifies to the practical value of our model.