The Grasshopper Problem on the Sphere

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.

Figures (23)

• Figure 1: Study of discretization effects. The difference between the discrete probability ${P}_h(\theta)$ of a hemispherical lawn for different spherical grid types and sizes, and the exact continuous grasshopper probability $p_h(\theta) = 1 - \theta/\pi$ (top panel) and the corresponding absolute difference on a logarithmic scale (middle panel). The discrete probabilities were obtained by averaging across a thousand random orientations of the hemispherical lawn relative to the grid. The width of the corresponding distribution (one standard deviation) is shown in the bottom panel. The discretization errors are smaller than $0.1\%$ for most values of $\theta$.
• Figure 2: Optimal lawn configurations for $\theta = 0.98 \pi$ on the $t$-design grid (left), the HEALPix grid (middle), and the Goldberg grid (right). All grids have on the order of 40,000 points. It can be seen that the underlying symmetry of the Goldberg grid heavily influences the resulting optimal lawn shape.
• Figure 3: Centered histograms of potential energies \ref{['eq:potentialenergy']} across the entire grid for $t$-design (left panel), HEALPix (middle panel), and Goldberg polyhedron grids (right panel) for the representative jump angle $\theta=0.30\pi$. The histograms for $t$-designs and HEALPix grids are more regular, with a width that decreases as the grid resolution $N$ is increased. The HEALPix grid histograms are more prominently peaked due to a higher concentration of points near the poles. The distributions for Goldberg polyhedron grids are much less regular with a larger overall width that does not appear to decrease with increasing $N$.
• Figure 4: Optimal grasshopper spin configurations in the antipodal one-lawn setup for different values of the jump $\theta$.
• Figure 5: Left: Number of cogs in numerically found optimal lawn configurations in the cogwheel regime. The data for the antipodal one-lawn setup (red empty squares) is compared to the corresponding data for the non-antipodal one-lawn setup (green filled squares) and for the antipodal two-lawn setup (blue empty diamonds). In antipodal setups the number of cogs must be odd, while for non-antipodal lawns the number of cogs can be any integer. Solid black lines correspond to multiples of $2\pi/\theta$ -- these modes are accessible in all setups. Dashed black lines show additional modes that are accessible only in the two-lawn setup, corresponding to odd multiples of $\pi/\theta$. Most numerically found optimal configurations belong to the lowest allowed mode. Right: Height of cogs in numerically found optimal lawn configurations in the antipodal one-lawn setup. Vertical dashed lines denote jump angles of the form $\theta=\theta_q=\pi/q$ for integer $q$.