CQIS: The Grasshopper Problem

一只蝗虫随机地落在给定区域的平坦草坪上。然后，它会以随机的方向跳跃一次，距离是固定的。草坪应该是什么形状，才能最大限度地增加蚱蜢在跳跃后留在草坪上的机会？答案远不是显而易见的！事实上，这个容易说但很难解决的数学问题与量子信息和统计物理都有意想不到的联系。球面上的广义版本可以提供对一类新的贝尔不等式的洞察，这是一种可通过实验验证的数学表达式，它捕捉到了量子力学所描述的世界与我们日常“经典”理解的一些不同之处。此外，这个问题的离散版本可以用来对自旋系统或微观磁铁系统进行建模，这种相互作用的方式可能会导致统计物理学中有趣的新结果。尽管深度出人意料，但蚱蜢问题可以很容易地理解，而不需要任何先前的物理知识，因此提供了一个很好的方法来让学生以及普通公众对统计物理和量子信息感兴趣。对于将致力于这一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，因为计算和分析工具广泛适用于不同的科学领域。拟议的研究的目标是使用分析和数值方法，包括模拟退火法和并行回火，探索蝗虫问题和相应的自旋系统的性质，重点是它们与涉及随机测量选择的贝尔不等式的联系。贝尔定理是量子物理中最基本的定理之一。然而，关于贝尔不等式的全部类别，即使是对两个自旋1/2粒子的自旋测量的最简单的情况，仍有许多有待发现。研究更一般的贝尔不等式可以加深我们的理解，即量子关联可以比使用经典模型的任何关联强得多。这提高了我们对基本量子物理的认识，也在贝尔实验、量子通信和量子密码学中有重要的应用，因为随机选择可能有助于使密码协议更安全和更有效。除此之外，相关的自旋系统代表了一类新的具有固定范围相互作用的统计模型，其中范围可以很大。这些模型展示了一系列不同寻常的特性，例如跳跃某些值的复杂、断开的“基态”自旋配置。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.