Nonlocal Setting Research Articles

Locally distinguishing genuinely nonlocal sets with entanglement resource

A set of orthogonal product states is deemed genuinely nonlocal if they remain locally indistinguishable under any bipartition. In this paper, we first construct a genuinely nonlocal product basis B I(5, 3) in C5⨂C5⨂C5 using a set of nonlocal product states in C3⨂C4 . Then, we obtain a genuinely nonlocal product basis B II(5, 3) by replacing certain states in B I(5, 3) with some superposition states. We achieve perfect discrimination of the constructed genuinely nonlocal product basis, separately employing two EPR states and one GHZ state. Our protocol is more efficient than quantum teleportation.

Genuinely nonlocal sets without entanglement in multipartite systems

Genuinely nonlocal sets without entanglement in multipartite systems

Strong quantum nonlocality and unextendibility without entanglement in N-partite systems with odd N

A set of orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. Although such a phenomenon has been shown to any three-, four-, and five-partite systems, the existence of strongly nonlocal orthogonal product sets in multipartite systems remains unknown. In this paper, by using a general decomposition of the N-dimensional hypercubes, we present strongly nonlocal orthogonal product sets in N-partite systems for all odd N≥3. Based on this decomposition, we give explicit constructions of unextendible product bases in N-partite systems for odd N≥3. Furthermore, we apply our results to quantum secret sharing, uncompletable product bases, and PPT entangled states.

Strongest nonlocal sets with minimum cardinality in tripartite systems

Strongest nonlocal sets with minimum cardinality in tripartite systems

Strongest nonlocal sets with small sizes

Strongest nonlocal sets with small sizes

Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection

We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution, with the same mass, of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a “projection” of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one.Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting.

Finite Element Discretizations for Variable-Order Fractional Diffusion Problems

We present a finite element discretization scheme for multidimensional fractional diffusion problems with spatially varying diffusivity and fractional order. We consider the symmetric integral form of these nonlocal equations defined on general geometries and in arbitrary bounded domains. A number of challenges are encountered when discretizing these equations. The first comes from the heterogeneous kernel singularity in the fractional integral operator. The second comes from the formally dense discrete operator with its quadratic growth in memory footprint and arithmetic operations. An additional challenge comes from the need to handle volume conditions—the generalization of classical local boundary conditions to the nonlocal setting. Satisfying these conditions requires that the effect of the whole domain, including both the interior and exterior regions, be computed on every interior point in the discretization. Performed directly, this would result in quadratic complexity. In order to address these challenges, we propose a strategy that decomposes the stiffness matrix into three components. The first is a sparse matrix that handles the singular near-field separately, and is computed by adapting singular quadrature techniques available for the homogeneous case to the case of spatially variable order. The second component handles the remaining smooth part of the near-field as well as the far-field, and is approximated by a hierarchical {mathcal {H}}^2 matrix that maintains linear complexity in storage and operations. The third component handles the effect of the global mesh at every node, and is written as a weighted mass matrix whose density is computed by a fast-multipole type method. The resulting algorithm has therefore overall linear space and time complexity. Analysis of the consistency of the stiffness matrix is provided and numerical experiments are conducted to illustrate the convergence and performance of the proposed algorithm.

Locally distinguishing nonlocal sets with entanglement resource

Recently, much attention has been paid to the construction of nonlocal sets. The existence of nonlocal sets gives rise to the research on the entanglement-assisted discrimination. We first present a method to locally distinguish a nonlocal set of orthogonal product states in (ℂd)⊗n by utilizing only a 2⊗2 maximally entangled state. Then we generalize the method to general multipartite quantum system ⊗i=1nℂdi based on the same entangled resource. Fortunately, we find that a 2⊗2 maximally entangled state can be employed as a universal resource for perfectly distinguishing the nonlocal sets constructed by Zhen et al (2022). In particular, the protocols we designed could not only efficiently distinguish these states, but also show that the strength of these nonlocal sets is minimal from the perspective of auxiliary resources.

Completable sets of orthogonal product states with minimal nonlocality

Since the theory of quantum nonlocality without entanglement was proposed, great achievements have been made in the local distinguishability of bipartite orthogonal quantum states, but there are still many unresolved problems. In this paper, we first propose the concept of minimal nonlocality to characterize the lower bound of the amount of the elements contained in a nonlocal set of orthogonal product states. A nonlocal set of orthogonal product states has minimal nonlocality if and only if the set can be distinguished by local operations and classical communication (LOCC) after only a certain state of the set is removed. Then we give a novel method to construct completable sets with minimal nonlocality in bipartite quantum system. Compared with the existing results, the completable sets we construct have the least number of elements in the same quantum systems. Finally, we extend our construction method of completable nonlocal set with fewer elements in bipartite system to tripartite or even more multipartite quantum systems.

Isomorphism of nonlocal sets of orthogonal product states in bipartite quantum systems

Isomorphism of nonlocal sets of orthogonal product states in bipartite quantum systems

Strong quantum nonlocality without entanglement in an n -partite system with even n

In multipartite systems, great progress has been made recently on the study of strong quantum nonlocality without entanglement. However, the existence of orthogonal product sets with strong quantum nonlocality in even party systems remains unknown. Here the even number is greater than four. In this paper, we successfully construct strongly nonlocal orthogonal product sets in $n$-partite systems for all even $n$, which answers the open questions given by Halder et al. [\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.040403} {Phys. Rev. Lett \textbf{122}, 040403 (2019)}] and Yuan et al. [\href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.042228} {Phys. Rev. A \textbf{102}, 042228 (2020)}] for any possible even party systems. Thus, we find general construction of strongly nonlocal orthogonal product sets in space $\otimes_{i=1}^{n}\mathcal{C}^{d_{i}}$ ($n,d_{i}\geq 3$) and show that there do exist incomplete orthogonal product bases that can be strongly nonlocal in any possible $n$-partite systems for all even $n$. Our newly constructed orthogonal product sets are asymmetric. We analyze the differences and connections between these sets and the known orthogonal product sets in odd party systems. In addition, we present a local state discrimination protocol for our sets by using additional entangled resource. When at least two subsystems have dimensions greater than three, the protocol consumes less entanglement than teleportation-based protocol. Strongly nonlocal set implies that the information cannot be completely accessed as long as it does not happen that all parties are together. As an application, we connect our sets with local information hiding in multipartite system.

Nonlocality without entanglement in general multipartite quantum systems

The construction of nonlocal sets of quantum states has attracted much attention in recent years. We first introduce two Lemmas related to the triviality of orthogonality-preserving local measurements. Then we propose a general construction of nonlocal set of $n(d-1)+1$ orthogonal product states in $(\mathbb{C}^{d})^{\otimes n}$. The sets of nonlocal orthogonal product states are also put forward for the multipartite quantum systems with arbitrary dimensions. Our novel construction gives rise to nonlocal sets of orthogonal product states with much less members and thus reveals the phenomenon of nonlocality without entanglement more efficiently.

Remarks on the Fractional Moser—Trudinger Inequality

In this article, we study the connection between the fractional Moser-Trudinger inequality and the fractional $\left(\frac{kp}{p-1},p\right)$-Poincar\'e type inequality for any Euclidean domain and discuss the sharpness of this inequality whose analogous results are well known in the local case. We further provide sufficient conditions on domains for fractional $(q,p)$-Poincar\'e type inequalities to hold. We also derive Adachi-Tanaka type inequalities in the non-local setting.

Large time behavior of solutions to a class of hyperbolic inequalities with mixed nonlinearities

We investigate the large time behavior of solutions to a class of inhomogeneous hyperbolic inequalities involving combined nonlinearities of the form |u|p+1Γ(σ)∫0t(t−s)σ−1|∇u(s,x)|qds+w(x), where p,q>1 and σ>0. We show that, if w has a positive average, then the considered class admits no global weak solution. Our approach is based on nonlinear capacity estimates specifically adapted to the nonlocal setting of our problem.

Fractional Hardy equations with critical and supercritical exponents

We study the existence, nonexistence and qualitative properties of the solutions to the problem (P)(-Δ)su-θu|x|2s=up-uqinRNu>0inRNu∈H˙s(RN)∩Lq+1(RN),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} ({\\mathcal {P}}) \\quad \\quad \\left\\{ \\begin{aligned} (-\\Delta )^s u -\ heta \\frac{u}{|x|^{2s}}&=u^p - u^q \\quad \ ext {in }\\,\\, {\\mathbb {R}}^N\\\\ u&> 0 \\quad \ ext {in }\\,\\, {\\mathbb {R}}^N\\\\ u&\\in {\\dot{H}}^s({\\mathbb {R}}^N)\\cap L^{q+1}({\\mathbb {R}}^N), \\end{aligned} \\right. \\end{aligned}$$\\end{document}where sin (0,1), N>2s, q>pge {(N+2s)}/{(N-2s)}, theta in (0, Lambda _{N,s}) and Lambda _{N,s} is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions, we mean both the radial symmetry, that is obtained by using the moving plane method in a nonlocal setting on the whole mathbb {R}^N, and a suitable upper bound behavior of the solutions. To this last end, we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space.

Propagation Phenomena for Nonlocal Dispersal Equations in Exterior Domains

This paper is concerned with the spatial propagation of bistable nonlocal dispersal equations in exterior domains. We first obtain the existence and uniqueness of an entire solution which behaves like a planar traveling wave front for large negative time. Then, when the entire solution comes to the interior domain, the profile of the front will be disturbed. However, the disturbance is local in space for finite time, which means the disturbance disappears as its location is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile uniformly in space and continue to propagate in the same direction for large positive time provided that the interior domain is compact and convex. Our work generalizes the local (Laplace) diffusion results obtained by Berestycki et al. (2009) to the nonlocal dispersal setting by using new known Liouville results and Lipschitz continuity of entire solutions due to Li et al. (2010).

Nonlocal sets of orthogonal product states with the less amount of elements in tripartite quantum systems

Nonlocal sets of orthogonal product states with the less amount of elements in tripartite quantum systems

Regularity theory for nonlocal equations with VMO coefficients

We prove higher regularity for nonlinear nonlocal equations with possibly discontinuous coefficients of VMO type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to obtain higher integrability, in our nonlocal setting we are able to also prove a substantial amount of higher differentiability, so that our result is in some sense of purely nonlocal type. By embedding, we also obtain higher Hölder regularity for such nonlocal equations.

Genuine hidden nonlocality without entanglement: from the perspective of local discrimination

Quantum nonlocality without entanglement is a fantastic phenomenon in quantum theory. This kind of quantum nonlocality is based on the task of local discrimination of quantum states. Recently, Bandyopadhyay and Halder (2021 Phys. Rev. A 104 L050201) studied the problem: is there any set of orthogonal states which can be locally distinguishable, but under some orthogonality preserving local measurement, each outcome will lead to a locally indistinguishable set. The set with such property is called to have hidden nonlocality. Moreover, if such phenomenon can not arise from discarding subsystems which is termed as local irredundancy, we call it genuine hidden nonlocality. There, they presented several sets of entangled states with genuine hidden nonlocality. However, they doubted the existence of a set without entanglement but with genuine hidden nonlocality. In this paper, we eliminate this doubt by constructing a series of sets without entanglement but whose nonlocality can be genuinely activated. We derive a method to tackle with the local irredundancy problem which is a key tricky for the systems whose local dimensions are composite numbers. Unexpectedly, the constructions of genuine hidden nonlocal sets without entanglement seems to be easier than that with entanglement. Therefore, from this perspective, this kind of nonlocality is rather different from the Bell nonlocality.

CIDER: An Expressive, Nonlocal Feature Set for Machine Learning Density Functionals with Exact Constraints.

Machine learning (ML) has recently gained attention as a means to develop more accurate exchange-correlation (XC) functionals for density functional theory, but functionals developed thus far need to be improved on several metrics, including accuracy, numerical stability, and transferability across chemical space. In this work, we introduce a set of nonlocal features of the density called the CIDER formalism, which we use to train a Gaussian process model for the exchange energy that obeys the critical uniform scaling rule for exchange. The resulting CIDER exchange functional is significantly more accurate than any semilocal functional tested here, and it has good transferability across main-group molecules. This work therefore serves as an initial step toward more accurate exchange functionals, and it also introduces useful techniques for developing robust, physics-informed XC models via ML.