AI 中文总结
研究经典与量子半参数模型的统计效率，通过统一处理推广相关界，引入基本概念并应用于多例子，以奇异值分解阐明信道效应，应用于光学成像问题，计算成像极限，展示空间模式解复用优势，有望成为经典和量子统计重要工具。
AI 中文摘要
在经典和量子统计中，高维未知参数众多，使用半参数模型对其做最小假设较为明智。本文对经典和量子半参数模型的统计效率进行统一处理，推广了克拉美 - 罗界和赫尔斯托姆界。先从抽象和几何角度引入基本概念并应用于诸多例子，深入探讨半参数效率理论中的信道，用奇异值分解阐明其统计效应。应用于相干和非相干光学成像问题，能计算成像分辨率的经典和量子极限。对于亚衍射非相干成像，空间模式解复用在估计广义傅里叶系数上优于直接成像且更接近量子极限。该理论有望成为经典和量子统计的重要工具，应用于传感和成像等领域。
英文摘要
In classical and quantum statistics, high-dimensional unknown parameters are abundant and it is often prudent to make minimal assumptions about them using so-called semiparametric models. To attack a wide range of semiparametric problems in one broad stroke, we present a unified treatment of statistical efficiency for classical and quantum semiparametric models, generalizing the Cramér-Rao and Helstrom bounds beyond finite-dimensional parameters. We introduce the fundamental concepts in abstract and geometric terms before applying them to many examples, covering general classical and quantum models as well as the paradigmatic special cases of Gaussian and Poisson fields. We give an in-depth treatment of channels in the semiparametric efficiency theory and advocate the use of the singular value decomposition to elucidate the statistical effects of channels. To demonstrate the utility of the formalism, we apply it to coherent and incoherent optical imaging problems, assuming an arbitrary field or intensity on the object plane without parametric assumptions. Our formalism enables us to compute classical and quantum limits to coherent and incoherent imaging resolution in statistical terms. For subdiffraction incoherent imaging, we demonstrate that spatial-mode demultiplexing can be far superior to direct imaging in estimating generalized Fourier coefficients and come closer to the quantum limits. We envision our theory becoming an essential tool for both classical and quantum statistics with useful applications to sensing and imaging, whenever minimal assumptions about a high-dimensional parameter should be made.
Comments85 pages, 19 figures. Comments welcome
