the bound for $\Phi=0$, so WVA achieves enhancement of the Fisher information. This Fisher information enhancement effect, which does not happen always, it has been observed for certain WVA schemes [@viza2013; @jordan2014].

There is no contradiction between the facts that the minimum probability of error, obtained by making use of the concept of trace distance, is not changed by WVA, while at the same time there can be enhancement of the Fisher information. By selecting a particular probability distribution to evaluate the Fisher information, we include information about the detection scheme. In our case, we estimate the value of $\tau$ by measuring the $\tau$-dependent shift of the centroid of the spectrum of the signal in one output port after PBS$_2$, which is only part of all the information available, given by the full signal in Eqs. (\[projections1\]) and (\[projections2\]). We also assumed a Gaussian probability distribution with a constant variance $\sigma^2$ independent of $\tau$. The Cramér-Rao bound we have derived here depends on the full information available (the quantum state) before any particular detection. An unitary transformation, as WVA is, does not modify the bound. On the contrary, the Fisher information, by using a particular probability distribution to describe the 