From Quantum Entanglement to Tensor Decomposition by Global Optimization

纠缠和可分性是一对孪生子。纠缠是表征系统内多个部分耦合或相互作用的最基本模式;可分性是以一种等价但更显着的关系来表示复杂系统，以便于理解和控制。本计画的目的是利用最佳化的新工具，发展出数位量测纠缠态与其最近可分态间“绝对”能隙的方法。初步目标是建立一个量子信息学背景下利用全局优化技术测量纠缠和可分性的基本范式。通过适度的修改，该范式可以应用于不同的领域。这项研究的结果将使解决许多其他背景下的可分性问题成为可能，例如经济发展，农业生产，工业制造，环境演变，社交网络和应用力学，其中成分，因素，部分或子系统经常相互交织。量子纠缠被认为是许多应用不可或缺的资源，因为量子计算的潜力可以快速，并行计算子系统之间的非线性相关性使得传统的分解技术难以分析。另一方面，张量的概念也获得了新的关注，由于其巨大的描述灵活性。这两种结构在纠缠和可分性方面具有相似的特征。在这两条战线上都开展了许多活动并取得了许多成就。然而，数值测量纠缠态与其最近的可分离态之间的“绝对”间隙的途径从未完全进行过。该项目旨在通过全局优化技术在一个框架下解决量子纠缠和低秩张量近似。当全局优化完成后，在规定的误差容限内，我们掌握了给定状态与可分离状态集之间的度量，通过该度量我们可以衡量纠缠的质量，得出给定系统是否鲁棒纠缠的结论，并将知识扩展到其他应用。该项目旨在建立理论和算法基础：1）利用纠缠的几何特性; 2）为新算法开发一个通用平台，有效地提高鲁棒性，速度和准确性; 3）探索推广到具有额外约束的应用。这项研究连同由此产生的软件包，预计将找到广泛的适用性，从量子力学扩展到数据分析，网络分析和其他领域。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。

Entanglement and separability are twins. Entanglement is the most basic mode when characterizing the coupling or interaction of multiple parts within a system; separability is to represent the complicated system in an equivalent but more manifesting relationship for understanding and control. This project aims to develop methods to numerically measure the "absolute" gap between an entangled state and its nearest separable state with the new tool of global optimization techniques. The initial goal is to establish a basal paradigm for gauging entanglement and separability with global optimization technologies in the context of quantum informatics. With modest modification, the paradigm can be applied across different fields. Results from this research will make it possible to address separability issues in many other contexts, such as economic development, agricultural production, industrial manufacture, environmental evolution, social networks, and applied mechanics, where constituents, factors, parts, or subsystems are regularly intertwined.Quantum entanglement is regarded as an indispensable resource for many applications due to the potential of quantum computing for fast, concurrent computation. The nonlinear correlations among subsystems make it difficult to analyze by traditional decomposition techniques. On the other hand, the notion of tensors has also gained new attention thanks to its great descriptive flexibility. Both structures share similar features concerning entanglement and separability. There have been many activities and achievements on both fronts. Yet, the avenue of numerically measuring the "absolute" gap between an entangled state and its nearest separable state has never been fully undertaken. This project aims to tackle both quantum entanglement and low-rank tensor approximation under one framework by global optimization techniques. When global optimization is finished, within the prescribed error tolerance we have in hand the metric between a given state and the set of separable states, by which we can gauge the quality of entanglement, draw conclusions on whether the given system is robustly entangled, and extend the knowledge to other applications. This project aims to establish theoretic and algorithmic foundations to: 1) exploit the geometric properties of entanglement; 2) develop a common platform for new algorithms effective in robustness, speed, and accuracy; and 3) explore the generalization to applications with additional constraints. This research together with the resulting software package is expected to find wide applicability extending from quantum mechanics to data analysis, network analysis, and other fields. The work will solidify study of many features under one unified framework.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.