\omega_0 \tau\right)
\right] \label{output_result},\end{aligned}$$ which is the same result \[see Eq. (\[input\_result\])\] obtained for the signal after PBS$_1$, but before PBS$_2$.

We can also see the previous results from a slightly different perspective making use of the Cramér-Rao inequality [@helstrom1976]. The WVA scheme considered throughout can be thought as a way of estimating the value of the single parameter $\tau$ with the help of a light pulse in a coherent state $|\alpha
\rangle$. Since the quantum state is pure, the minimum variance that can show any unbiased estimation of the parameter $\tau$, the Cramér-Rao inequality, reads $$\mathrm{Var} \left( \hat{\tau} \right) \ge \frac{1}{4}\left[ \langle
\frac{\partial \alpha}{\partial \tau}| \frac{\partial
\alpha}{\partial \tau} \rangle-\left| \langle \alpha |
\frac{\partial \alpha}{\partial \tau}\rangle\right|^2
\right]^{-1}, \label{cramer1}$$ Making use of Eq. (\[state1\]), one obtains that here the Cramér-Rao inequality reads [@derivative] $$\mathrm{Var} \left( \hat{\tau} \right) \ge \frac{1}{2N \left(
\omega_0^2+B^2\right)} \label{cramer2}$$ where $B=\sqrt{2 \ln 2}/T_0$ is the rms bandwidth in angular frequency of the pulse. In all cases of interest $B \ll
\omega_0$. The Cramér-Rao inequality is a fundamental limit that set a bound to the minimum variance that any measurement can achieve. It is unchanged by unitary transformations and only depends on the quantum state considered.

Inspection of Eqs. 