Grasshopper problem yields insight into quantum theory

December 4, 2017 by Lisa Zyga feature

 (Phys.org)—Like many mathematical puzzles, the grasshopper problem is simple to state but difficult to solve: A grasshopper lands at a random point on a lawn of area 1, then jumps once, a fixed distance, in a random direction. What shape should the lawn be in order to maximize the chance that the grasshopper stays on the lawn after jumping?

A first impression may be that the lawn should be in the shape of a circle, at least when the distance the grasshopper jumps is small. However, Olga Goulko and Adrian Kent, the two physicists who introduced the grasshopper problem in a new paper, have mathematically proved that a disc-shaped lawn is not optimal for any distance.

Instead, they discovered through numerical simulations that the optimal lawn shape takes on a variety of complex shapes for different jumping distances, such as a cogwheel shape for distances smaller than 1/π1/2(the radius of a circle of area 1, or approximately 0.56), while for larger distances, the optimal lawn consists of disconnected pieces. Often, but not always, these optimal shapes possess some type of symmetry.

Motivated by physics

Aside from being an interesting geometry problem, the grasshopper problem is also closely related to research in quantum physics and may have a variety of technological applications. In particular, the grasshopper problem is connected to the Bell inequalities, which famously show that, unlike classical physics models, quantum theory does not obey local realism. A prime example of the violation of local realism is seen in quantum entanglement, in which two distant entangled systems exhibit correlations that cannot be explained by any model that obeys local realism.

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