量子高斯传感中的 James-Stein 估计
James-Stein Estimation in Quantum Gaussian Sensing
AI总结：
本文将 James-Stein 估计引入量子计量，基于量子高斯传感分析纠缠、相干与噪声对其相较 MLE 优势的影响，并用于多变量后选择计量以降低实验误差。
摘要：
James-Stein 估计量是一种有偏估计量——在样本数有限时，其期望值并不等于真实均值。最大似然估计量（MLE）是无偏且渐近最优的。然而，在估计 3 个或更多正态分布随机变量的均值时，James-Stein 估计量的总体（期望）误差小于 MLE。我们从频率学派和贝叶斯学派两种视角，将 James-Stein 估计量引入量子计量学领域。我们借助量子高斯传感这一视角，刻画量子现象对 James-Stein 估计量的影响；量子高斯传感的任务是估计未知多变量量子高斯态的均值。我们发现，无噪声的纠缠或相干性能够提升 James-Stein 估计量的性能，但会削弱其相对于 MLE 的优势。在存在噪声的情况下，James-Stein 优势得以恢复。量子效应也可以增强 James-Stein 优势。我们通过研究多变量后选择计量（广义弱值放大）证明了这一点，该策略利用量子效应，通过不完美探测器测量参数。仅仅通过以不同方式对测量数据进行后处理，我们的技术就能降低量子实验中的误差。
英文摘要：
The James-Stein estimator is a biased estimator -- for a finite number of samples its expected value is not the true mean. The maximum-likelihood estimator (MLE), is unbiased and asymptotically optimal. Yet, when estimating the mean of $3$ or more normally-distributed random variables, the James-Stein estimator has a smaller total (expected) error than the MLE. We introduce the James-Stein estimator to the field of quantum metrology, from both the frequentist and Bayesian perspectives. We characterise the effect of quantum phenomena on the James-Stein estimator through the lens of quantum Gaussian sensing, the task of estimating the mean of an unknown multivariate quantum Gaussian state. We find that noiseless entanglement or coherence improves performance of the James-Stein estimator, but diminishes its advantage over the MLE. In the presence of noise, the James-Stein advantage is restored. Quantum effects can also boost the James-Stein advantage. We demonstrate this by investigating multivariate postselective metrology (generalised weak-value amplification), a strategy that uses quantum effects to measure parameters with imperfect detectors. Simply by post-processing measured data differently, our techniques reduce errors in quantum experiments.
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