The Grasshopper Problem on the Sphere

• URL: http://arxiv.org/abs/2603.08579v1
• Date: Mon, 09 Mar 2026 16:34:45 GMT
• Title: The Grasshopper Problem on the Sphere
• Abstract summary: The spherical grasshopper problem is a geometric optimization problem that arises in the context of Bell inequalities.<n>We examine the role of spherical discretization and compare three natural variants of the problem.<n>We analyze the geometric structure of the corresponding optimal lawn configurations and interpret it in terms of a spherical harmonics expansion.
• Score: 1.1635287453977645
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The spherical grasshopper problem is a geometric optimization problem that arises in the context of Bell inequalities and can be interpreted as identifying the best local hidden variable approximation to quantum singlet correlations for measurements along random axes separated by a fixed angle. In a parallel publication [arXiv:2504.20953], we presented numerical solutions for this problem and explained their significance for singlet simulation and testing. In this companion paper, we describe in detail the geometric and computational framework underlying these results. We examine the role of spherical discretization and compare three natural variants of the problem: antipodal complementary lawns, antipodal independent lawns, and non-antipodal complementary lawns. We analyze the geometric structure of the corresponding optimal lawn configurations and interpret it in terms of a spherical harmonics expansion. We also discuss connections to other physical models and to classical problems in geometric probability.

Related papers

• The Shape of Addition: Geometric Structures of Arithmetic in Large Language Models [11.106770083374288]
  Large Language Models exhibit paradoxical hallucination in fundamental arithmetic, implying a disconnect between internal computation and discrete output.<n>We propose the Noisy Quantization Model to explain this geometry, framing arithmetic errors as Geometric Slippages caused by internal neural noise pushing a continuous, latent Carry Potential across quantization thresholds.<n>We validate these insights through a geometric consistency check method that effectively detects and corrects these quantization failures during inference.
  arXiv  Detail & Related papers  (2026-05-29T10:55:50Z)
• Visual Diffusion Models are Geometric Solvers [54.31602846693932]
  We show that visual diffusion models can serve as effective geometric solvers by working in pixel space.<n>We first demonstrate this on the Inscribed Square Problem, a long-standing problem in geometry.<n>We extend the approach to two other well-known hard geometric problems: the Steiner Tree Problem and the Simple Polygon Problem.
  arXiv  Detail & Related papers  (2025-10-24T17:57:31Z)
• Topological Obstructions and How to Avoid Them [22.45861345237023]
  We show that local optima can arise due to singularities or an incorrect degree or winding number. We propose a new flow-based model that maps data points to multimodal distributions over geometric spaces.
  arXiv  Detail & Related papers  (2023-12-12T18:56:14Z)
• Geometric Neural Diffusion Processes [55.891428654434634]
  We extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling. We show that with these conditions, the generative functional model admits the same symmetry.
  arXiv  Detail & Related papers  (2023-07-11T16:51:38Z)
• Geometrical, topological and dynamical description of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model and their interplay with quantum entanglement [0.0]
  This work investigates the connections between integrable quantum systems with quantum phenomena exploitable in quantum information tasks. We find the relevant dynamics, identify the corresponding quantum phase space and derive the associated Fubini-Study metric. By narrowing the system to a two spin-$mathtts$ system, we explore the relevant entanglement from two different perspectives.
  arXiv  Detail & Related papers  (2022-10-29T11:53:14Z)
• Squashed entanglement in one-dimensional quantum matter [0.0]
  We show that edge squashed entanglement discriminates unambiguously between topological insulators and topological superconductors. Such topological squashed entanglement is robust under variations of the sample conditions due to disorder or local perturbations. We show that topological squashed entanglement defines the natural measure of nonlocal correlation patterns in quantum matter.
  arXiv  Detail & Related papers  (2022-07-27T21:40:53Z)
• Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents [102.42623636238399]
  We identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. We derive algorithms that exploit these geometric structures to solve these problems efficiently.
  arXiv  Detail & Related papers  (2022-03-20T16:23:17Z)
• Projections of Orbital Measures and Quantum Marginal Problems [0.0]
  We prove integral formulae for the probability densities, establish some properties of the densities, and discuss connections to problems in representation theory. As applications, we show a number of results on marginal problems in quantum information theory and also prove an integral formula for restriction multiplicities.
  arXiv  Detail & Related papers  (2021-12-27T21:21:22Z)
• Geometric phase in a dissipative Jaynes-Cummings model: theoretical explanation for resonance robustness [68.8204255655161]
  We compute the geometric phases acquired in both unitary and dissipative Jaynes-Cummings models. In the dissipative model, the non-unitary effects arise from the outflow of photons through the cavity walls. We show the geometric phase is robust, exhibiting a vanishing correction under a non-unitary evolution.
  arXiv  Detail & Related papers  (2021-10-27T15:27:54Z)
• Hilbert-space geometry of random-matrix eigenstates [55.41644538483948]
  We discuss the Hilbert-space geometry of eigenstates of parameter-dependent random-matrix ensembles. Our results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We compare our results to numerical simulations of random-matrix ensembles as well as electrons in a random magnetic field.
  arXiv  Detail & Related papers  (2020-11-06T19:00:07Z)
• Geometry of Similarity Comparisons [51.552779977889045]
  We show that the ordinal capacity of a space form is related to its dimension and the sign of its curvature. More importantly, we show that the statistical behavior of the ordinal spread random variables defined on a similarity graph can be used to identify its underlying space form.
  arXiv  Detail & Related papers  (2020-06-17T13:37:42Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.