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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_00025v2 (pdf) |
| Date submitted: | 2023-09-18 11:27 |
| 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 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 two referees report a positive judgment of our work. The two referees raise several interesting questions and point out different technical points to clarify. We have taken into account, in the new version of the manuscript, the different comments of the referees.
In the reply of each report, we have included a revised version of the manuscript with changes written with colors. We hope that these comments and clarifications will render this article suitable for publication in SciPost Physics.
Yours sincerely,
the authors
List of changes
- p 2. New reference.
- p 13. New sentence: "we recall that
the equato ris the set of states maximizing the increase of QFI"
- p 16-17. New paragraph : "This is due to... pole of the Bloch sphere".
- p 17. Modification of a sentence: "We observethat..."
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 2023-10-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202306_00025v2, delivered 2023-10-07, doi: 10.21468/SciPost.Report.7908
Report
I am a bit confused by the author's response to the first comment. On one hand, the authors disagree with the comment, and on the other hand, it is stated that "The QFI is reached only when the difference between the two density matrices goes to zero", which agrees with the comment. Do the authors mean although they use orthogonality as the objective it is never achieved and the distance of the states remains small? This does not agree with the examples of the manuscript where the distance is not necessarily infinitesimally small.
In practice, it is fine to use the finite difference as a heuristic cost function for the optimization, but it should be kept in mind that this can differ from the QFI. It is fine to write the finite difference as $F_{fd}(t_f) =\frac{8}{\delta X^2}$ but assume it equals to the QFI and write $F=\frac{8}{\delta X^2}$, as if the QFI depends on $\delta X$, is confusing. Actually, the authors have observed the differences as the Bures distance and the QFI do not coincide with each other in some examples in the manuscript.
For the third point on why the other pole is not chosen as the other objective state, the authors did not fully address the question: "Even the steered state cannot reach it, why it is not chosen so the steered state can be made as close to it as possible?". An objective state does not necessarily have to be reached, it just sets a target that steers the state as close to it as possible. It is not quite clear why the orthogonal states cease to be good target states when they can not be reached.
Author: Quentin Ansel on 2023-10-27 [id 4071]
(in reply to Report 2 on 2023-10-07)We thank the Referee for these interesting comments.
1) We almost agree with the opinion of the Referee but there are some confusions, which may arise from our previous reply. We therefore start the explanation from the beginning to clarify the different points mentioned by the Referee.
Consider the finite difference version of the QFI: $\mathcal F_{fd} = 4 D/\delta X^2$. This quantity strictly reaches the QFI when $\delta X$ tends to zero. Now, suppose that for some $\delta X$, it is possible to freely set the value of D (i.e., we have complete controllability of the system). Then we can choose D^2=2 which corresponds to the maximum value of D, and we have $ F_{fd} = 8/\delta X^2$. However, the controls that allow us to generate orthogonal states have an increasing duration when \delta X decreases. Consequently, for $\delta X$ -> 0, the control time goes to infinity, and in practice, we cannot produce orthogonal states in a finite time for two infinitesimally close values of X.
We can reverse the reasoning. We can set a finite value of $F$, and search for a value of $\delta X$ such that $F_{fd}$ is very close to $F$. For $\delta X$ small enough, we are sure that such a situation can happen. Thus, in specific situations, we can expect that both QFI and selective optimizations lead to the same (or at least, similar) control process, and the result should be almost identical with both approaches. This is basically the message given in pages 8 and 9 of the manuscript. Note that in Eq.~(17), we have written $\mathcal F \simeq \frac{8}{\delta X^2} $ and not $\mathcal F= \frac{8}{\delta X^2}$, to specify that the quantities are not strictly equivalent.
However, there are some very specific cases in which $\mathcal F_{fd} =\mathcal F$ for a non-zero value of $\delta X$. 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 (see examples Sec. 5.3 and 5.4). Such a situation can be described as follows: Set a value $\mathcal F$ that must be obtained at time $t$, choose $\delta X$ such that $\mathcal F= \frac{8}{\delta X^2}$, and find the control that generates $D^2 = 2$ in time $t$. Of course, this reasoning must be taken with caution because the problem may not have a solution. We present a counter-example in Fig. 4 where the Bures distance and the QFI do not agree very well. We are in this case outside the domain of validity in which $\mathcal F_{fd}$ is very close to $\mathcal F$.
2) We agree with the second comment of the Referee: " An objective state does not necessarily have to be reached; it just sets a target that steers the state as close to it as possible". We could have chosen orthogonal states for this optimization procedure. It is likely that, in this situation, the result would not have been very different. However, since we know that the maximum of the cost functional cannot be reached, the result may be suboptimal with respect to the goal of increasing the distance between the two systems. We therefore prefer in this case not to use orthogonal states. Numerical simulations show that the South Pole is a good choice of target state.
We modified the text to clarify these points. We have attached a version of the manuscript with changes in red characters.
Attachment:
Article_QFI_sélectivité_v2-1.pdf