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Quantumenhanced multiparameter estimation and compressed sensing of a field
by Youcef Baamara, Manuel Gessner, Alice Sinatra
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Authors (as registered SciPost users):  Youcef Baamara · Manuel Gessner 
Submission information  

Preprint Link:  scipost_202207_00046v1 (pdf) 
Date submitted:  20220728 18:02 
Submitted by:  Baamara, Youcef 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We show that a significant quantum gain corresponding to squeezed or oversqueezed spin states can be obtained in multiparameter estimation by measuring the Hadamard coefficients of a 1D or 2D signal. The physical platform we consider consists of twolevel atoms in an optical lattice in a squeezedMott configuration, or more generally by correlated spins distributed in spatially separated modes. Our protocol requires the possibility to locally flip the spins, but relies on collective measurements. We give examples of applications to scalar or vector field mapping and compressed sensing.
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Reports on this Submission
Report #1 by Jing Liu (Referee 1) on 2022828 (Invited Report)
 Cite as: Jing Liu, Report on arXiv:scipost_202207_00046v1, delivered 20220828, doi: 10.21468/SciPost.Report.5572
Report
Quantum multiparameter estimation is a core problem in quantum parameter estimation, and the performance of a quantum system on multiparameter estimation could dramatically different with the singleparameter counterpart. Hence, the study of multiparameter estimation in various quantum systems or scenarios and design of schemes that present quantum advantages are very essential. The authors study a multiparameter estimation problem in a collective spin system, and provide a scheme for the measurement of the linear combinations of a set of parameters. The paper is interesting and would benefit the scientists working on both quantum parameter estimation and collective spin systems. I think I can recommend it to be published on SciPost. Hereby are my suggestions and comments.
1. The entire analysis in this paper is based on the measurement of an observable, and the corresponding transfer function is up to first order. In quantum parameter estimation, a useful tool to analyze the theoretical performance of the multiparameter estimation is the quantum CramerRao bound [more details could be found in a recent review: J. Phys. A: Math. Theor. 53 023001 (2020)]. Of course this tool has some drawbacks like it could be unattainable, but the comparison between it and Eqs. (6) and (13) would give the readers some impressions how good the provided scheme is. I suggest the authors calculate the quantum Fisher information matrix in this case and do some comparison.
2. In the provided scheme, the parameters are not measured directly, but calculated via linear combinations. As a matter of fact, the performance of different types of linear combinations is different. From the perspective of quantum CramerRao bound, the quantum Fisher information matrix of linear combinations and the original parameters are connected via the Jacobian matrix. Hence, I am curious that whether there exists a set of $epsilon_k$ that gives the minimum total variance of the original parameters. This analysis may also tell the readers if the measurement of Hadamard coefficients is optimal in theory.
3. The discussion of quantum imaging in Sec. 4 is very interesting. Yet, there are not many technical details like how to calculate the dynamics numerically or analytically in Eq. (25) in the case of L_H=512*512. The total Hilbert space seems very large in this case. I think the presentation of more technical details would reduce the difficulty of repeating the results in the paper, and help the readers to better understand the corresponding meaning.
Author: Youcef Baamara on 20220928 [id 2862]
(in reply to Report 1 by Jing Liu on 20220828)
Answer to question 1:
Indeed, the quantum Fisher information matrix is important for evaluating the quality of the metrological gain obtained in a multiparameter estimation scheme. We added subsection B.1 in Appendix B where we calculate the quantum Fisher information matrix. Its maximum eigenvalue, which gives the optimal quantum gain that could be achieved by the OAT states, is then compared in figure 4 with the metrological gain obtained by our strategy in the absence and presence of decoherence.
Answer to question 2:
In subsection B.1 of appendix B we obtained that for a state generated by the OAT dynamics, the quantum Fisher information matrix admits a single combination of parameters that achieves a quantum advantage. Thus, for a given state, only one Hadamard coefficient can be obtained with a metrological gain. In order to obtain another coefficient we need to change the state via the local flips of the individual spins as shown in the text. A change of the $\epsilon_k$ to select a given Hadamard coefficient is achieved via a transformation that leaves the spectrum of the quantum Fisher information matrix invariant.
Answer to question 3:
The details of the calculation of the squeezing dynamics described by equation (25) are presented in a previous paper. We have added a sentence at the end of section 4 where we make a precise reference to the formulas that we use from that reference.
Author: Youcef Baamara on 20220928 [id 2861]
(in reply to Report 2 on 20220904)Answer to questions 1 and 2:
In multiparameter estimation theory, it is in general the matter of estimating N parameters by means of repeated measurements of $N$ observables of the system starting from the same quantum state (the observables can possibly be measured simultaneously in one realization of the experiment in case they commute). In this framework, the covariance matrix of the estimators and the quantum Fisher matrix $\cal F$, which are related by the multiparameter CramÃ©rRao inequality, are usually introduced. However, as we now show it in the appendix B.1, for a oneaxistwisting squeezed state it is not possible to obtain a quantum gain in the estimation of each parameter by this strategy. For a given quantum state, only one eigenvalue of $\cal F$, corresponding to a particular combination of the parameters shows a quantum advantage when all the others show a disadvantage. To obtain a quantum advantage in another combination it is necessary to change the state. To estimate $N$ combinations with a quantum advantage, $N^2$ measurements should then be performed. In our strategy on the contrary, for a given state only one collective observable must be measured to estimate the combination that shows a quantum advantage. To estimate the $N$ combinations, it is sufficient to measure $N$ collective observables.
Answer to question 3:
Analytical results concerning the metrological gain and its scaling with $N$ are discussed in detail in an earlier article. To point this out, we have added a sentence at the end of section 2 that shows the reader where to find the technical details.
Answer to question 4:
A sentence was added in the introduction to cite previous works in multiparameter estimation and a footnote to explain the considered frame and results of the cited works and to show the advantage of our strategy.