From Quantum Entanglement to Tensor Decomposition by Global Optimization

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.