CQIS: The Grasshopper Problem

University Of Massachusetts Boston, Dorchester MA

Investigators

Abstract

A grasshopper lands at a random point on a flat lawn of given area. It then jumps once, a fixed distance, in a random direction. What shape should the lawn be to maximize the chance that the grasshopper remains on the lawn after jumping? The answer turns out to be far from obvious! In fact, this easily stated yet hard to solve mathematical problem has unexpected connections to both quantum information and statistical physics. A generalized version on the sphere can provide insight into a new class of Bell inequalities, which are experimentally verifiable mathematical expressions that capture some of the ways in which the world described by quantum mechanics differs from our everyday “classical” understanding. Additionally, a discrete version of the problem can be used to model a system of spins, or microscopic magnets, interacting in ways that may lead to interesting new results in statistical physics. Despite this unexpected depth, the grasshopper problem can be easily understood without any prior physics knowledge, and hence offers a great way to get students, as well as the general public, interested in statistical physics and quantum information. For the students, both graduate and undergraduate, who will be working on the problem, it will also be a perfect introduction to computational techniques for physical models, as the algorithms and existing codes are simple to use and to build on. The project will thus contribute to training the future STEM workforce, as the computational and analytical tools are broadly applicable in different scientific fields. The goal of the proposed research is to explore the properties of the grasshopper problem and the corresponding spin system using analytical and numerical methods, including simulated annealing and parallel tempering, with focus on their connection to Bell inequalities that involve random measurement choices. Bell's theorem is one of the most fundamental theorems in quantum physics. However, much still remains to be discovered about the full class of Bell inequalities, even for the simplest case of spin measurements on two spin 1/2 particles. Studying more general Bell inequalities can deepen our understanding of how much stronger quantum correlations can be than any correlations possible using classical models. This advances our knowledge of fundamental quantum physics and also has important applications to Bell experiments, quantum communication, and quantum cryptography, as random choices may help make cryptographic protocols safer and more efficient. Besides this, the associated spin system represents a new class of statistical models with fixed-range interactions, where the range can be large. These models exhibit an array of unusual properties, such as complex disconnected "ground state" spin configurations for certain values of the jump. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.