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arXiv:2512.14302 · Quantum Science and Technology

Universal Structure of Nonlocal Operators for Deterministic Navigation and Geometric Locking

Jia Bao, Bin Guo, Shu Qu, Fanqin Xu, Zhaoyu Sun

quant-phSubmitted Dec 16, 2025• Announced Dec 17, 2025• 1 citation
v1 · Dec 16, 2025v2 · Apr 24, 2026
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Abstract

We establish a universal geometric framework that transforms the search for optimal nonlocal operators from a combinatorial black box into a deterministic predict-verify operation. We discover that the principal eigenvalue governing nonlocality is rigorously dictated by a low-dimensional manifold parameterized by merely two fundamental angular variables, $θ$ and $φ$, whose symmetry leads to further simplification. This geometric distillation establishes a precise mapping connecting external control parameters directly to optimal measurement configurations. Crucially, a comparative analysis of the geometric angles against the principal eigenvalue spectrum, including its magnitude, susceptibility, and nonlocal gap, reveals a fundamental dichotomy in quantum criticality. While transitions involving symmetry sector rotation manifest as geometric criticality with drastic operator reorientation, transitions dominated by strong anisotropy exhibit geometric locking, where the optimal basis remains robust despite clear signatures of phase transitions in the spectral indicators. This distinction offers a novel structural classification of quantum phase transitions and provides a precision navigation chart for Bell experiments.

DOI
10.1088/2058-9565/ae8a78

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We establish a universal geometric framework that transforms the search for optimal nonlocal operators from a combinatorial black box into a deterministic predict-verify operation. We discover that the principal eigenvalue governing nonlocality is rigorously dictated by a low-dimensional manifold parameterized by merely two fundamental angular variables, θθθ and φφφ, whose symmetry leads to further simplification. This geometric distillation establishes a precise mapping connecting external control parameters directly to optimal measurement configurations. Crucially, a comparative analysis of the geometric angles against the principal eigenvalue spectrum, including its magnitude, susceptibility, and nonlocal gap, reveals a fundamental dichotomy in quantum criticality. While transitions involving symmetry sector rotation manifest as geometric criticality with drastic operator reorientation, transitions dominated by strong anisotropy exhibit geometric locking, where the optimal basis remains robust despite clear signatures of phase transitions in the spectral indicators. This distinction offers a novel structural classification of quantum phase transitions and provides a precision navigation chart for Bell experiments.
Multipartite nonlocality is a measure of multipartite quantum correlations, and has been used to characterize quantum criticality in various low-dimensional quantum models. In this work, we propose the nonlocality spectrum ${|{\ensuremath{\lambda}}_{i}|}$ of the nonlocality transfer matrix as a tool to characterize multipartite nonlocality and quantum criticality in infinite-size translation-invariant quantum systems. The approach is used to analyze four typical quantum spin chain models, i.e., the transverse-field Ising model, an XXZ model, a three-body coupling cluster model and an XYZ model. In various types of quantum phase transitions in these models, it is found that at the critical points, the leading eigenvalue $|{\ensuremath{\lambda}}_{1}|$ presents identifiable signals, and the spectrum gap $\mathrm{\ensuremath{\Delta}}|\ensuremath{\lambda}|=|{\ensuremath{\lambda}}_{1}|\ensuremath{-}|{\ensuremath{\lambda}}_{2}|$ closes. Moreover, the nonlocality spectrum provides a unified mathematical framework to reveal the underlying mechanisms for various previously reported behaviors of multipartite nonlocality. Our results demonstrate the power of the nonlocality spectrum as a valuable tool for the comprehensive characterization of multipartite nonlocality and quantum criticality in one-dimensional quantum systems.
Based on transfer matrix theory and the multipartite nonlocality spectrum [Formula: see text], we systematically investigate ground-state nonlocal correlations across three canonical infinite, translation-invariant one-dimensional spin models, including the XXZ model, the three-body coupled Cluster–Ising model, and the Kitaev ladder model, which cover both conventional and topological Quantum Phase Transitions (QPTs). We show that the emergence of multipartite nonlocal correlations can be quantitatively determined via the leading eigenvalue criterion [Formula: see text], offering a concise and reliable indicator for diagnosing nonlocal quantum features. Further numerical comparisons reveal that the leading eigenvalue [Formula: see text] is strongly model-dependent and lacks universal applicability for identifying quantum critical behaviors across different QPT types. In contrast, the nonlocality spectral gap [Formula: see text] exhibits prominent robustness against variations in system symmetry, coupling strength, and intrinsic transition mechanisms. It characterizes quantum criticality through three distinct response modes, thereby enabling a unified description of both conventional and topological QPTs. This work establishes the nonlocality spectrum as a versatile and powerful framework for the joint characterization of multipartite nonlocality and quantum criticality in one-dimensional many-body quantum systems.
In this paper, we characterize the global multipartite nonlocality in one-dimensional infinite-size spin-1/2 chains. For this purpose, we calculate the multipartite nonlocality of nnn-site subchains, and take the large-nnn limit. As nnn increases from 2 to 100, we find that the average revenue and the marginal revenue of nonlocality converge gradually, thus they can be used to characterize the global multipartite nonlocality in infinite-size chains. Furthermore, in the vicinity of the quantum phase transition (QPT) points of the transverse-field Ising model and the XXZ\mathit{XXZ}XXZ model, in the large-nnn limit, multipartite nonlocality becomes extremely sensitive to external perturbation, that is, arbitrarily small perturbation away from the QPT points would lead to a drastic change in the global nonlocality. This signal of QPTs is easily distinguishable, and can be used to detect QPTs. We also draw a vivid physical picture of multipartite correlations and QPTs by analogy with magnetic domains in ferromagnets.
In quantum lattice models, in the large-NNN limit, boundary conditions have little effect upon local observables for sites in the centers of the lattices. In this paper, we will study the boundary effects upon multipartite nonlocality (a kind of multipartite quantum correlation associated with Bell-type inequalities) in one-dimensional finite-size spin chains, both for zero temperature and for finite temperatures. We define a quantity $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ to characterize the boundary effects, where S\mathcal{S}S is a measure of global multipartite nonlocality of the entire lattice, and $\ensuremath{\delta}\mathcal{S}$ is the difference of the measure induced by changing the boundary conditions. We find $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ does not vanish in the large-NNN limit. Instead, at zero temperature, with the increase of NNN, $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ would increase steadily in the vicinity of the quantum phase transition point of the models, and converge to a nonzero constant in noncritical regions. It shows clearly that boundary effects generally exist, in the form of multipartite correlations, in long chains. The boundary effects are explained by the competition between the two orders of the models. In addition, based on these numerical results, we construct a Bell inequality, which is violated by chains with periodic (closed) boundary conditions and not violated by chains with open boundary conditions. Furthermore, we study $\frac{\ensuremath{\delta}{\mathcal{S}}_{T}}{{\mathcal{S}}_{T}}$ of finite-size chains at finite temperatures, and show that boundary effects survive in finite temperature regions.
Multipartite Bell nonlocality provides a device-independent probe of many-body quantum correlations, but its characterization is limited by the rapid growth of the underlying classical and quantum optimization problems. We develop a scalable method for constructing and certifying permutationally invariant Bell inequalities using only one- and two-body correlators. The construction gives families of inequalities with robust quantum violations for general mmm measurements as the number of parties NNN becomes large. To improve robustness against noise, we optimize the ratio of the quantum value to the classical bound for these families in the large-NNN limit. We then certify the resulting quantum violation using semidefinite programming. For the broad class of Bell inequalities studied here, the infinite-NNN ratios take simple rational values for finite mmm and converge to coth⁡(1)\coth(1)coth(1) as m→∞m\to\inftym→∞. The optimized inequalities efficiently detect many-body Bell nonlocality with collective measurements, with more measurement settings leading to stronger violations.
A problem in quantum information theory is to find the experimental setup that maximizes the nonlocality of correlations with respect to some suitable measure such as the violation of Bell inequalities. There are however some complications with Bell inequalities. First and foremost it is unfeasible to determine the whole set of Bell inequalities already for a few measurements and thus unfeasible to find the experimental setup maximizing their violation. Second, the Bell violation suffers from an ambiguity stemming from the choice of the normalization of the Bell coefficients. An alternative measure of nonlocality with a direct information-theoretic interpretation is the minimal amount of classical communication required for simulating nonlocal correlations. In the case of many instances simulated in parallel, the minimal communication cost per instance is called nonlocal capacity, and its computation can be reduced to a convex-optimization problem. This quantity can be computed for a higher number of measurements and turns out to be useful for finding the optimal experimental setup. Focusing on the bipartite case, we present a simple method for maximizing the nonlocal capacity over a given configuration space and, in particular, over a set of possible measurements, yielding the corresponding optimal setup. Furthermore, we show that there is a functional relationship between Bell violation and nonlocal capacity. The method is illustrated with numerical tests and compared with the maximization of the violation of CGLMP-type Bell inequalities on the basis of entangled two-qubit as well as two-qutrit states. Remarkably, the anomaly of nonlocality displayed by qutrits turns out to be even stronger if the nonlocal capacity is employed as a measure of nonlocality.