The Grasshopper Problem on the Surface of the Sphere

A unit sphere is covered by a lawn such that, for every pair of antipodal points, exactly one point belongs to the lawn. A grasshopper lands randomly on the lawn and then jumps in a random direction through a fixed spherical angle θ. The task is to find the optimal shape of the lawn that maximizes the probability that, after jumping, the grasshopper will successfully land back on the lawn. Surprisingly, this mathematical problem is closely linked to statistical physics and quantum information theory, specifically Bell’s inequalities.

In the quantum context, two parties measure spins of an entangled singlet state along randomly chosen axes separated by a fixed angle θ. The lawn corresponds to a Local Hidden Variable Model (LHVM) for the given angle θ. Optimizing the grasshopper’s success probability yields LHVMs that exhibit maximal anticorrelations, and thus in a natural sense represent the best possible classical simulation of the singlet quantum state. In this talk, we present and characterize numerically obtained optimal lawn shapes for all values 0 < θ < 𝜋. Our results give a rich new quantitative characterization of the Bell non-locality of the singlet as a resource in quantum cryptography and other applications.