f)_i \right\}$, for a given $\tau$, is known. If we assume that the probability distribution $p(\left\{(\Delta f)_i
\right\} |\tau)$ is Gaussian, with mean value $\Delta f$ given by Eq. (\[centroid\_shift\]) and variance $\sigma^2$, determined by the errors inherent to the detection process, the Fisher information reads [@fisher1] $$I(\tau)=\frac{N}{\sigma^2} \left[ \frac{\partial \Delta
f}{\partial \tau}\right]^2$$ where $$\frac{\partial \Delta f}{\partial \tau}=\frac{\gamma B^2 \left[
B^2 \tau^2 \sin \phi-\omega_0 \tau \left( \gamma+\cos
\phi\right)-\sin \phi \left( 1+\gamma \cos \phi\right)
\right]}{2\pi\, \left( 1+\gamma \cos \phi\right)^2}$$ and $\phi=\omega_0 \tau-\Gamma$.

For $\phi=0$, i.e., the angle of post-selection is $\theta=-\pi/2+\omega_0 \tau$, the Fisher information is $$I_0=\frac{N_0}{2} \left( 1+\gamma\right) \times \frac{\gamma^2 B^4
(\omega_0 \tau)^2}{2\pi\sigma^2 (1+\gamma)^2}= \frac{\gamma^2 B^4
(\omega_0 \tau)^2}{4\pi\sigma^2 (1+\gamma)} \label{fisher_bound1}$$

Notice that $\theta=-\pi/2$ corresponds to considering equal input and output polarization state, i.e., no weak value amplification scheme. For $\phi=\pi$, where the angle of post-selection is $\theta=\pi/2+\omega_0 \tau$, we have $$I_{\pi}=\frac{N_0}{2} \left( 1-\gamma\right) \times \frac{\gamma^2
B^4 (\omega_0 \tau)^2}{2\pi\sigma^2 (1-\gamma)^2}= \frac{\gamma^2
B^4 (\omega_0 \tau)^2}{4\pi\sigma^2 (1-\gamma)}
\label{fisher_bound2}$$ $\theta=\pi/2$ corresponds to considering an output polarisation state orthogonal to the input polarisation state i.e., when the effect of weak value amplification is most dramatic, as it can be easily observed in Fig. 2(a). The Fisher bound for $\Phi=\pi$ is a factor $I_{\pi}/I_0=(1+\gamma)/(1-\gamma)$ larger than 