From Quantum Entanglement to Tensor Decomposition by Global Optimization

纠缠和可分离是孪生兄弟。纠缠是刻画系统内多个部分的耦合或相互作用时最基本的模式；可分离性是将复杂的系统表现为一种等价但更明显的关系，以便于理解和控制。该项目旨在开发利用全局优化技术的新工具来数值测量纠缠态与其最近的可分离态之间的“绝对”间隙的方法。最初的目标是在量子信息学的背景下，建立一个利用全局优化技术测量纠缠和可分离性的基本范式。只要稍加修改，这一范例就可以应用于不同的领域。这项研究的结果将有可能解决许多其他背景下的可分性问题，如经济发展、农业生产、工业制造、环境进化、社会网络和应用力学，这些领域中的成分、因素、部分或子系统规则地相互交织。由于量子计算在快速、并行计算方面的潜力，量子纠缠被认为是许多应用中不可或缺的资源。各子系统之间的非线性相关性使传统的分解方法难以进行分析。另一方面，张量的概念也因其巨大的描述性灵活性而获得了新的关注。这两种结构在纠缠和可分离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，取得了许多成果。然而，用数值方法测量纠缠态与其最近的可分离态之间的“绝对”间隙的方法从未被完全采用。该项目旨在通过全局优化技术在一个框架下同时处理量子纠缠和低阶张量近似。当全局优化完成后，在规定的误差容限内，我们可以获得给定态和可分离态集合之间的度量，通过它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，以：1)开发纠缠的几何特性；2)为新算法开发一个通用平台，有效地提高健壮性、速度和精度；3)探索对附加约束应用的推广。这项研究和由此产生的软件包预计将发现从量子力学到数据分析、网络分析和其他领域的广泛适用性。这项工作将在一个统一的框架下巩固对许多特征的研究。该奖项反映了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。

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.