Globe-hopping

Abstract

We consider versions of the grasshopper problem (Goulko & Kent 2017 Proc. R. Soc. A 473, 20170494) on the circle and the sphere, which are relevant to Bell inequalities. For a circle of circumference 2π, we show that for unconstrained lawns of any length and arbitrary jump lengths, the supremum of the probability for the grasshopper's jump to stay on the lawn is one. For antipodal lawns, which by definition contain precisely one of each pair of opposite points and have length π, we show this is true except when the jump length ϕ is of the form π(p/q) with p, q coprime and p odd. For these jump lengths, we show the optimal probability is 1 - 1/q and construct optimal lawns. For a pair of antipodal lawns, we show that the optimal probability of jumping from one onto the other is 1 - 1/q for p, q coprime, p odd and q even, and one in all other cases. For an antipodal lawn on the sphere, it is known (Kent & Pitalúa-García 2014 Phys. Rev. A 90, 062124) that if ϕ = π/q, where $q\in N$ , then the optimal retention probability of 1 - 1/q for the grasshopper's jump is provided by a hemispherical lawn. We show that in all other cases where 0 < ϕ < π/2, hemispherical lawns are not optimal, disproving the hemispherical colouring maximality hypotheses (Kent & Pitalúa-García 2014 Phys. Rev. A 90, 062124). We discuss the implications for Bell experiments and related cryptographic tests.

Keywords: Bell inequalities; geometric combinatorics; grasshopper problem.

Figure 1.

Sketch of an optimal general lawn S_L,q with L = π/3, for a rational jump ϕ = (p/q)2π with p = 2 and q = 5 (a) and of an optimal antipodal lawn S_π,q for a rational jump ϕ = π(p/q) with even numerator p = 2 and q = 5 (b). (Online version in colour.)

Figure 2.

Sketch of an optimal antipodal lawn S′_q for a rational jump ϕ = π(p/q) with odd numerator p = 1 (a), p = 3 (b), p = 5 (c), and (in all panels) q = 8. The two demi-lawns are marked by different line styles (black solid lines and red dashed lines). (Online version in colour.)

Figure 3.

Construction of modified lawns for rational ϕ = pπ/q and p even, with p = 6 and q = 13 (a, lemma 3.8) and for irrational ϕ/π with ϕ = 1.2 (b, lemma 3.9). The caps and cups around the equator are shown as black and white dots, respectively.

Figure 4.

Spherical triangle (a) and spherical coordinates (b). (Online version in colour.)

Figure 5.

Jump geometry.