量子多参数自适应贝叶斯估计及其在超分辨成像中的应用
Quantum Multi-Parameter Adaptive Bayesian Estimation and Application to Super-Resolution Imaging
摘要：
在贝叶斯估计理论中，估计器${\hat θ} = E[θ|l]$获得最小均方误差（MMSE），用于在给定$θ$上的先验$P_θ$的情况下，从通过噪声信道$P_{l|θ}$的$l$的观测来估计感兴趣的标量参数$θ$。在量子感测任务中，用户获得$ρ_θ$，即编码$θ$的量子状态。它们选择测量，正算子值测量（POVM）$Π_l$，其将信道$P_{l|θ} = {\rm Tr}(ρ_θΠ_l)$诱导到测量结果$l$，在其上采用前述经典MMSE估计器。Personick发现了最佳POVM $Π_l$，它在所有可能的测量中最小化MMSE，并且MMSE。1971年的这个结果不如量子费雪信息（QFI）广为人知，当$P_θ$不可用时，量子费雪信息降低了所有测量的无偏估计量的方差。对于多参数估计，即，当$θ$是向量时，在Fisher量子估计理论中，QFI矩阵的逆提供了无偏估计器的协方差的算子下界。然而，对于{ \em贝叶斯}设置中的多参数量子估计，在量化量子限制和测量设计方面的工作很少。在本文中，我们建立在Personick的结果，构建一个贝叶斯自适应测量计划的多参数估计时，$N$副本的$ρ_θ$。我们说明了一个应用程序，以本地化一个集群的点发射器在一个高度分瑞利角视场，在荧光显微镜和天文学的一个重要问题。我们的算法转化为一个多空间模式转换之前的光子检测阵列，与电光反馈，以适应模式分选。我们表明，这种接收机的性能远远优于量子噪声限制的焦平面直接成像。
英文摘要：
In Bayesian estimation theory, the estimator ${\hat θ} = E[θ|l]$ attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest $θ$ from the observation of $l$ through a noisy channel $P_{l|θ}$, given a prior $P_θ$ on $θ$. In quantum sensing tasks, the user gets $ρ_θ$, the quantum state that encodes $θ$. They choose a measurement, a positive-operator valued measure (POVM) $Π_l$, which induces the channel $P_{l|θ} = {\rm Tr}(ρ_θΠ_l)$ to the measurement outcome $l$, on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM $Π_l$ that minimizes the MMSE over all possible measurements, and that MMSE. This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements, when $P_θ$ is unavailable. For multi-parameter estimation, i.e., when $θ$ is a vector, in Fisher quantum estimation theory, the inverse of the QFI matrix provides an operator lower bound to the covariance of an unbiased estimator. However, there has been little work on quantifying quantum limits and measurement designs, for multi-parameter quantum estimation in the {\em Bayesian} setting. In this paper, we build upon Personick's result to construct a Bayesian adaptive measurement scheme for multi-parameter estimation when $N$ copies of $ρ_θ$ are available. We illustrate an application to localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs far superior to quantum-noise-limited focal-plane direct imaging.
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