We propose a metrological strategy reaching Heisenberg-scaling precision in the estimation of functions of any fixed number $p$ of arbitrary parameters encoded in a generic $M$-channel linear network. This scheme is experimentally feasible since it only employs a single-mode squeezed vacuum and homodyne detection on a single output channel. Two auxiliary linear networks are required and their role is twofold: to refocus the signal into a single channel after the interaction with the interferometer, and to fix the function of the parameters to be estimated according to the linear network analyzed. Although the refocusing requires some knowledge on the parameters, we show that the required precision on the prior measurement is achievable with a classic measurement. We conclude by discussing two paradigmatic schemes in which the choice of the auxiliary stages allows us to change the function of the unknown parameter to estimate.

中文翻译：

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线性光网络中参数函数估计中的海森堡标度精度

我们提出了一种计量策略，在估计任何固定数的函数时达到海森堡标度精度 $p$ 以泛型编码的任意参数 $米$-通道线性网络。该方案在实验上是可行的，因为它仅在单个输出通道上采用单模压缩真空和零差检测。需要两个辅助线性网络，它们的作用是双重的：在与干涉仪交互后将信号重新聚焦到单个通道中，并根据分析的线性网络确定待估计参数的函数。尽管重新聚焦需要有关参数的一些知识，但我们表明，通过经典测量可以实现先前测量所需的精度。我们最后讨论了两种范式方案，其中辅助阶段的选择允许我们改变未知参数的函数来估计。