CQIS: The Grasshopper Problem

一只蚱蜢落在给定面积的平坦草坪上的随机点上。然后，它跳一次，一个固定的距离，在一个随机的方向。草坪应该是什么形状，以最大限度地提高蚱蜢跳后留在草坪上的机会？答案其实并不明显！事实上，这个简单明了却难以解决的数学问题与量子信息和统计物理有着意想不到的联系。球上的广义版本可以提供对一类新的贝尔不等式的洞察，这些不等式是实验上可验证的数学表达式，它们捕获了量子力学所描述的世界与我们日常“经典”理解不同的一些方式。 此外，该问题的离散版本可以用于模拟自旋或微观磁体的系统，这些系统以可能导致统计物理学中有趣的新结果的方式相互作用。 尽管有这种意想不到的深度，蚱蜢问题可以很容易地理解，没有任何先前的物理知识，因此提供了一个很好的方式来让学生，以及公众，对统计物理和量子信息感兴趣。对于将致力于解决该问题的研究生和本科生来说，这也将是对物理模型计算技术的完美介绍，因为算法和现有代码易于使用和构建。因此，该项目将有助于培训未来的STEM劳动力，由于计算和分析工具广泛应用于不同的科学领域。拟议研究的目标是探索蚱蜢问题的性质和使用分析和数值方法，包括模拟退火和并行回火，重点是它们与涉及随机测量选择的Bell不等式的联系。贝尔定理是量子物理学中最基本的定理之一。然而，关于完整的贝尔不等式，即使是对两个自旋为1/2的粒子的自旋测量的最简单的情况，仍有许多东西有待发现。研究更一般的贝尔不等式可以加深我们对量子相关性比任何使用经典模型的相关性强多少的理解。这推进了我们对基础量子物理学的了解，并且在贝尔实验、量子通信和量子密码学中也有重要的应用，因为随机选择可能有助于使密码协议更安全、更有效。 除此之外，相关的自旋系统代表了一类新的统计模型与固定范围的相互作用，其中的范围可以很大。这些模型展示了一系列不寻常的特性，例如对于某些跳跃值，复杂的断开“基态”自旋配置。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.