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The grasshopper problem

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If you have a question about this talk, please contact Adrian Kent.

A grasshopper lands at a random point on a planar lawn of area one. It then makes one jump of fixed distance d in a random direction. What shape should the lawn be to maximize the chance that the grasshopper remains on the lawn after jumping? This easily stated yet hard to solve mathematical problem has intriguing connections to quantum information and statistical physics. A generalized version, where the lawn is placed on a unit sphere such that exactly one of every pair of antipodal points belongs to the lawn, provides insight into a new class of Bell inequalities that are relevant to quantum cryptography. In this setup two parties measure spins about randomly chosen axes and obtain correlations for pairs of axes separated by a fixed angle. A discrete version of the planar system can be represented by a generalized Ising model, where spins do not interact with their nearest neighbors but rather with spins a fixed distance away, and this distance d can be large. We show that, perhaps surprisingly, in two dimensions there is no d > 0 for which a disc shaped lawn is optimal. If the jump distance is smaller than the radius of the unit disc, the optimal lawn resembles a cogwheel, with transitions to more complex, disconnected shapes at larger d. A similar picture emerges for the spherical version of the problem. In this talk, we will discuss analytical and numerical results for the grasshopper problem in two and three dimensional flat space and on the surface of the sphere, as well as their connection to Bell’s inequalities involving random measurement choices.

This talk is part of the CQIF Seminar series.

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