SciPost Submission Page
Quantumenhanced multiparameter estimation and compressed sensing of a field
by Youcef Baamara, Manuel Gessner, Alice Sinatra
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Youcef Baamara · Manuel Gessner 
Submission information  

Preprint Link:  scipost_202207_00046v2 (pdf) 
Date submitted:  20220928 15:40 
Submitted by:  Baamara, Youcef 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

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.
Author comments upon resubmission
List of changes
 New affiliation for one of the authors (M.G.)
 A sentence was added in the introduction to cite previous works in multiparameter estimation (to which we added the references [12], [13] and [19]) and a footnote to explain the considered frame and results of the cited works and to show the advantage of our strategy.
 In the end of section 2, we added a sentence explaining where to find the technical details on the metrological gain and its scaling with N that were discussed in an earlier article.
 In subsection 3.2 we make a reference to the work of Goldberg [17].
 In the end of section 4, we added a sentence where we make a precise reference to the formulas needed to the calculation of the squeezing dynamics in the presence of decoherence described by equation (25).
 M.G. added funding informations to acknowledgments.
 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.
Current status:
Reports on this Submission
Report #3 by Jing Liu (Referee 1) on 2022107 (Invited Report)
Report
I am convinced by the authors' reply and revised manuscript and I recommend it to be published on SciPost now.
Report #1 by Anonymous (Referee 3) on 2022101 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202207_00046v2, delivered 20221001, doi: 10.21468/SciPost.Report.5808
Report
I am a bit puzzled by the authors' response on the advantage compared to the local measurement. It seems we may not be talking about the same local strategy. Take the case of estimating the scaler field at N sites, $\theta=(\theta_1, ...,\theta_N)$ ($\theta_j$ is at site j) for example, the local strategy is to prepare a squeezed state with N spins at one site(for example, N spins at site j) and estimate one parameter(for example $\theta_j$) at one time, then repeating the strategy at different sites to get the estimation of all N parameters. For the estimation of $\theta_j$ at site j, this is just a singleparameter problem and we have only quantum Fisher information, not quantum Fisher information matrix. So it is not clear how the eigenvalues of quantum Fisher information matrix come into play in the local strategy for the case of scaler field. Considering the sensitivity, for the local strategy with N spins prepared as the squeezed state at site j, the precision scales as $1/N^2$ for the estimation of $\theta_j$, and this has to be repeated N times to get all N parameters. For the collective strategy with N spins the precision also scales as $1/N^2$ for the estimation of one Hardmard coefficient and this also has to be repated N times to get all N Hardmard coeffi cients. There does not seem to be a difference on the precision.
I will appreciate if the authors can make a further clarification on the advantage.
Author: Youcef Baamara on 20221012 [id 2918]
(in reply to Report 1 on 20221001)1Framework 1.1 Our Strategy: In our work, we consider a configuration in which $N$ atoms distributed in spatially separated modes, for example at the nodes of an optical lattice (1 atom per site), are used to estimate $N$ parameters that are for example the values taken by a spatially extended field. As a result, the use of a coherent spin state (CSS), with $N$ collective measurements, allows us to estimate all the parameters with a variance (for one repetition of the measurement $\mu=1$) \begin{align} (\Delta\theta_k)_{\rm CSS}=1\qquad\forall k. \end{align} Quantum correlations, generated by the one axis twisting (OAT) dynamics, with $N$ collective measurements, allow for a quantum enhancement where all the parameters are estimated with variance (for one repetition of the measurement $\mu=1$) \begin{align} (\Delta\theta_k)=\xi(N,t)\qquad\forall k. \end{align} $\xi(N,t)$ is the squeezing parameter associated to the state generated by OAT at time $t$ for single parameter estimation.
1.2 The " Scanning Microscope " strategy: The "local strategy" mentioned in your report, is the one that we called in the introduction "Scanning Microscope", where one would use a set of $N$ atoms to locally estimate the field at each site. In this strategy, a coherent spin state, with collective measurements, allows us to estimate all the parameters with a variance (for one repetition of the measurement $\mu=1$) \begin{align} (\Delta\theta_k)_{\rm CSS}=\frac{1}{\sqrt N}\qquad\forall k. \end{align} Quantum correlations, generated by the one axis twisting (OAT) dynamics, with $N$ collective measurements, allow for a quantum enhancement where one obtains \begin{align} (\Delta\theta_k)=\frac{\xi(N,t)}{\sqrt N}\qquad\forall k. \end{align}
2 Comparison of the two strategies
3 Further modifications of the paper As a further change, we have expanded the paragraph in the introduction where we compare to the "Scanning microscope" approach to fully clarify this point.