concept of trace distance to look for the minimum probability of error achievable in [*any*]{} measurement when using a given quantum state. In doing that, we only considered how the quantum state changes for different values of the variable to be measured, without any consideration of how this quantum state is going to be detected. If we would like to include in the analysis additional characteristics of the detection scheme, one can use the concept of Fisher information, that requires to consider the probability distribution of possible experimental outcomes for a given value of the variable of interest. In this approach, one chooses different probability distributions to describe formally [*characteristics*]{} of specific detection scheme [@jordan2014].

Let us assume that to estimate the value of the delay $\tau$, we measure the shift of the centroid ($\Delta f$) of the spectrum $\Phi_u(\tau)$, given by Eq. (\[projections2\]). A particular detection scheme will obtain a set of results $\left\{ (\Delta
f)_i \right\}$, $i=1..N$ for a given delay $\tau$. $N$ is the number of photons detected. The Fisher information $I(\tau)$ provides a bound of $\mathrm{Var}\left( \hat{\tau} \right)$ for any unbiased estimator when the probability distribution $p(\left\{ (\Delta f)_i \right\}|\tau)$ of obtaining the set $\left\{ (\Delta 