Optimal Control Strategies for Parameter Estimation of Quantum Systems

We thank the Referee for this interesting comment. We agree that it is important to clarify the different quantities used in the paper. Following the comments, the text has been modified to highlight the differences between $\mathcal F$ and $\mathcal F_{fd}$. In particular, the comment in the footnote on page 9 is now inserted in the main text as follows:
$$\begin{equation} \mathcal F \simeq \frac{8}{\delta X^2} = 8 \alpha t_{\textrm{min}}^2, \end{equation}$$
with $\alpha$ a constant specific to the system. When $\delta X \rightarrow 0$, we obtain $t_{\textrm{min}} \rightarrow \infty$ and $\mathcal F \rightarrow \infty$. This result shall be manipulated with caution because $\mathcal F_{fd}$ gives us only an approximation of $\mathcal F$. However, in some cases, the optimization of the two quantities can lead to the same result. This is the case when the optimization process which generates orthogonal states for a non-zero value of $\delta X$ coincides with the optimal control of the QFI.

With this new formulation, we emphasize that $\mathcal F_{fd}$ is not necessarily equal to $\mathcal F$, but the optimization of $\mathcal F_{fd}$ can help maximizing $\mathcal F$. This point can be justified qualitatively from a Taylor expansion of the function and an exact treatment of the reminder. Consider for instance a function $f$ such that $f(0) = df/dx(0) =0$ (this is the case for the QFI). Then,
$$\exists c \in ]0,b[~|~\frac{d^2f}{dx^2}(c) = \frac{f(b)}{2 b^2}$$
Note the strict equality between the second derivative of the function at x=c and the function itself at x=b. Here $d^2f/dx^2(0)$ and $f(b)$ play respectively the role of $\mathcal F$ and $\mathcal F_{fd}$. For a fixed value of $b$, chosen small enough, we deduce that maximizing $f(b)$ also amounts to maximizing $d^2f/dx^2(c)$. If the variations of the second derivative $d^2f/dx^2$ are not too strong in the small interval $[0,c]$, this also amount to maximizing $d^2f/dx^2(c)$. The same kind of argument can be used in the paper for the QFI. This analysis in terms of Taylor expansion is not discussed in depth in the manuscript because it requires careful treatment of the logarithmic derivative operator $L$ (to be mathematically well justified). A brief footnote comment has been added in page 9, to clarify this point.