物理测量中的Wirtinger flow相位恢复与应用

In recent years, breakthrough progress of the compressed sensing (CS) theory in signal sampling shows its strong vitality. In our previous fund "compressed sensing theory and application in quantum physics", our research group has been successfully studied in the relation of L1 norm and quantum theory, imaging of single photon counting through CS, application of Phaselift in the calculation of fidelity and density matrix for multiphoton entanglement, obtained good agreement with the experimental results. In 2015, Prof. Candes proposed another more effective theory--Wirtinger flow (W-F), which is fast, stable, and universial in phase recovery. This project intends to extend the W-F theory to the fields of quantum physics and physical measurement. We will carry out phase recovery of different diffraction pattern and research of imaging in high resolution, and will combine W-F with the C-S theory to research on the diffraction of very weak light to obtain its intensity information and phase recovery. We will try to promote these results further to the X-ray diffraction and transmission electron diffraction, and develop them to an applied theory based on quantum theory, and provide new ideas and new methods for the study of quantum physics. This project is a research topic in the forefront of cross-cutting areas, and involved math, physics, and analysis measurement. It is strong exploratory, also practical application, and has potential significant both in theory and application.

近年来压缩感知(CS)理论在信号采样的突破性进展，展现出很强生命力。课题组在即将结题的面上基金“量子物理中的压缩感知理论与应用”中成功研究了L1范数与量子理论关联性、CS在单光子计数成像中应用、Phaselift在多光子纠缠保真度和密度矩阵计算中应用，获得了与实验相符的成果。2015年，Candès提出了更有效的Wirtinger flow(W-F)理论，在相位恢复方面更具快速、稳定和普适特性。本课题拟将W-F理论推广到量子物理与物理测量领域，对不同类型衍射像开展相位恢复及高分辨成像的研究，结合CS理论，研究在光子计数条件下的极弱光衍射像的获取和相位恢复，力图把这些成果推广到X光衍射和透射电子衍射，发展成量子理论基础上的应用理论，为量子物理研究提供新思路和新方法。本项目是前沿交叉领域研究课题，涉及数学、物理、分析测量多个学科，探索性强，又能实际应用，在理论和应用两方面均有重要意义。

本课题将新型高效的W-F数学理论推广到量子物理与物理测量领域，对不同类型衍射成像开展了相位恢复及高分辨成像的研究，结合CS理论，研究在光子计数条件下的极弱光衍射像的获取和相位恢复。主要成果包括：1).通过高密度波矢空间采样，实现了对复数光场的单幅弗朗禾费衍射的相位恢复；通过使用低自卷积的非中心对称单连通光阑构建光路，提高了相位恢复算法的稳定性与抗噪性，实现了在高噪声实验条件下的复数光场的单幅弗朗禾费衍射的相位恢复。通过采用低自卷积的掩模矩阵调制光场，构建了具有高抗噪性、抗过曝光性的相位恢复算法。2).利用phaselift模型结合Nesterov 算法重建多比特量子态密度矩阵借助 Phaselift 模型，引入Nestrov 快速算法数值模拟模拟重建了多比特量子态系统的密度矩阵，结果表明，由于phaselift 是基于压缩感知提出的估算模型，直接减少了实验测量数量，后期借助快速算法Nestrov，重建精度高、耗时较其他算法明显缩短；用更少基底重建多比特纯态量子态密度矩阵借助 IBM 平台，理论研究了如何用最少的泡利测量恢复重建密度矩阵，我们我们分别用泡利基底、以及子空间投影算子作为测量算子，恢复重建了不同比特 GHZ，W 态，Cluster 态的密度矩阵，发现对于纯态密度矩阵，只需要用两个方向的测量基底即可高精度恢复出密度矩阵。