We propose a measurement setup reaching Heisenberg scaling precision for the estimation of any distributed parameter $\phi$ (not necessarily a phase) encoded into a generic $M$-port linear network composed only of passive elements. The scheme proposed can be easily implemented from an experimental point of view since it employs only Gaussian states and Gaussian measurements. Due to the complete generality of the estimation problem considered, it was predicted that one would need to carry out an adaptive procedure which involves both the input states employed and the measurement performed at the output; we show that this is not necessary: Heisenberg scaling precision is still achievable by only adapting a single stage. The nonadapted stage only affects the value of a prefactor multiplying the Heisenberg scaling precision: We show that, for large values of $M$ and a random (unbiased) choice of the nonadapted stage, this prefactor takes a typical value which can be controlled through the encoding of the parameter $\phi$ into the linear network.

中文翻译：

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海森堡定标精度在多模量子计量学中的典型性

我们提出了一种可达到Heisenberg缩放精度的测量设置，可用于估计任何分布式参数 $\phi$ （不一定是阶段）编码为通用 $\mathrm{中号}$端口线性网络仅由无源元件组成。从方案的角度来看，所提出的方案很容易实现，因为它仅采用高斯状态和高斯测量。由于所考虑的估计问题的完全普遍性，可以预见的是，人们将需要执行一种自适应过程，该过程既要涉及所采用的输入状态，也要涉及对输出进行的测量。我们证明这不是必需的：仅通过调整单个阶段，仍然可以实现Heisenberg缩放比例精度。不适应的阶段只会影响乘以Heisenberg缩放精度的预因子的值：我们证明，对于$\mathrm{中号}$ 以及非适应阶段的随机（无偏）选择，该前置因子取一个典型值，该值可以通过参数编码来控制 $\phi$ 进入线性网络。