CQIS: The Grasshopper Problem

一只蚱蜢落在给定区域平坦草坪上的随机点上。然后它在一个随机的方向上跳跃一次，固定的距离。草坪应该是什么形状，以最大限度地提高蚱蜢跳跃后留在草坪上的机会？事实证明，答案远不是显而易见的！事实上，这个容易表述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。球体上的广义版本可以提供对一类新的贝尔不等式的见解，这些不等式是实验可验证的数学表达式，它们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模拟一个自旋系统，或者微观磁体，以某种方式相互作用，可能会导致统计物理学中有趣的新结果。尽管这个意想不到的深度，蚱蜢问题可以很容易地理解，没有任何先前的物理知识，因此提供了一个伟大的方式让学生，以及公众，对统计物理和量子信息感兴趣。对于将要研究这个问题的研究生和本科生来说，它也是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。由于计算和分析工具广泛适用于不同的科学领域，因此该项目将有助于培养未来的STEM劳动力。本研究的目的是利用分析和数值方法，包括模拟退火和平行回火，探索蚱蜢问题及其相应自旋系统的性质，重点研究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学中最基本的定理之一。然而，对于整类贝尔不等式，即使是对两个自旋为1/2的粒子的最简单的自旋测量，也还有很多有待发现的地方。研究更一般的贝尔不等式可以加深我们对量子相关性比使用经典模型的任何相关性强得多的理解。这提高了我们对基础量子物理的认识，也对贝尔实验、量子通信和量子密码学有重要的应用，因为随机选择可能有助于使加密协议更安全、更有效。除此之外，相关的自旋系统代表了一类具有固定范围相互作用的新统计模型，其中范围可以很大。这些模型表现出一系列不寻常的特性，比如对于某些跳跃值，复杂的断开的“基态”自旋配置。该奖项反映了美国国家科学基金会的法定使命，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.