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Optimal Control Strategies for Parameter Estimation of Quantum Systems
by Quentin Ansel, Etienne Dionis, Dominique Sugny
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Submission summary
| Authors (as registered SciPost users): | Quentin Ansel |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202306_00025v3 (pdf) |
| Date submitted: | 2023-10-27 14:06 |
| Submitted by: | Ansel, Quentin |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Optimal control theory is an effective tool to improve parameter estimation of quantum systems. Different methods can be employed for the design of the control protocol. They can be based either on Quantum Fischer Information (QFI) maximization or selective control processes. We describe the similarities, differences, and advantages of these two approaches. A detailed comparative study is presented for estimating the parameters of a spin$-\tfrac{1}{2}$ system coupled to a bosonic bath. We show that the control mechanisms are generally equivalent, except when the decoherence is not negligible or when the experimental setup is not adapted to the QFI. In this latter case, the precision achieved with selective controls can be several orders of magnitude better than that given by the QFI.
Author comments upon resubmission
Please find herewith a second revised version of the manuscript entitled ``Optimal control strategies for parameter estimation of quantum systems" that we would like to resubmit for publication in SciPost Physics
The first Referee has accepted the publication of this manuscript. Additional questions are raised by the second Referee. You can in our reply the responses to these new comments raised by the second Referee.
We hope that these comments and clarifications will render this article suitable for publication in SciPost Physics Core. We have also corrected some misprints that we have detected in the text.
Yours sincerely,
the authors
List of changes
List of changes :
-page 9 : « (orthogonal states may be generated only in infinite time for two infinitesimally close parameters) »
-page 9 : footnote, « $\Mc F_{fd}$ gives us only an approximation of $\Mc F$, but in some situations, the optimization of the two quantities can lead to the exact same result. This is the case when the optimization process that generates orthogonal states for a non-zero value of $\delta X$ coincides with the optimal control of the QFI. Several examples of such a situation are given in Sec.5. »
- page 17 : « These target states are not orthogonal, but they avoid the search for suboptimal solutions with respect to the goal of increasing the distance between the two systems. »
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 5) on 2023-11-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202306_00025v3, delivered 2023-11-03, doi: 10.21468/SciPost.Report.8042
Report
I appreciate the authors' clarification but further clarifications on the relationships between $F_fd$ and F may needed to avoid potential confusions:
1) It is right that making $\delta x$ 'sufficiently small' $F_{fd}$ can approach $F$. But 'sufficiently small' typically means the distance between the two states needs to be sufficiently small, i.e., D is sufficiently small. Since F corresponds to the second order expansion and typically $\delta x t$ goes into the expansion, so $\delta x t$ together needs to be suffiently small to make the second order expansion valid.
2)For the procedure that first choosing a \delta x such that $F_{target}=8/\delta x^2$ then find a control to make $D^2=2$, this procedure makes $F_{fd}=F_{target}$, but does not make $F_{fd}$ equal to the real $F$, since $F_{target}$ is just an arbitrary number set before the control and the final state, it does not equals to the QFI of the actual final state---unless $F_{fd}=F$ which goes back to the original point .
As I said, it is fine to use $F_{fd}$ as the figure of merit, but the authors should avoid possible confusions.
Author: Quentin Ansel on 2023-11-23 [id 4143]
(in reply to Report 1 on 2023-11-03)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.