Breaking the weak Heisenberg limit

We provide a very simple case showing that the weak form of the Heisenberg limit can be beaten while the prior information is improved without bias.

Figure 1

Posterior distribution $P\left(\tilde{\varphi }\right|\varphi )$ for the estimator $\tilde{\varphi }$ for $W={10}^{-3}, \overline{n}=1, m={10}^6, \nu =0.1$, and $\varphi ={10}^{-4}$. The exact and approximate expressions are indistinguishable. We have marked with vertical dashed lines the values of $\varphi$ and $W$.

Figure 2

Posterior distribution $P\left(\tilde{\varphi }\right|\varphi )$ for the estimator $\tilde{\varphi }$ for $W={10}^{-3}, \overline{n}=1, m=1.6\times {10}^4, \nu =0.03$, and $\varphi ={10}^{-4}$. The exact (solid line and approximate (dashed line) expressions are indistinguishable. We have marked with vertical dashed lines the values of $\varphi$ and $W$.