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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_00046v3 (pdf) 
Date submitted:  20221027 13:57 
Submitted by:  Baamara, Youcef 
Submitted to:  SciPost Physics 
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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 Anonymous (Referee 6) on 20221028 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202207_00046v3, delivered 20221028, doi: 10.21468/SciPost.Report.6001
Report
I appreciate the authors' detailed response. The authors has now mentioned that the strategy can not achieve high precision than the local strategy. But I am also not sure the strategy gives up the $1/\sqrt{N}$ scaling as the authors now state. It seems during the comparison the authors consider the local operator $s_j,x$ and the global $S_x$ in the same way. They are, however, different. The global operator has larger gap between the maximal and minimal energy level, thus higher precisions can be achieved for the estimation of its coefficient.
Another advantage the authors claimed is that "ours has the advantage that a single collective measurement has to be performed...all
measurements in our protocol are collective". It is not quite clear what the advantage is here. Collective measurement is in general harder to perform, 'all measurements are collective' does not sounds like an advantage. Do the authors mean "all measurements are the same"?
Author: Youcef Baamara on 20221103 [id 2980]
(in reply to Report 1 on 20221028)1 Indeed, the referee is right « the collective observable has larger gap between the maximal and minimal eigenvalue », and this is already taken into account in our calculation of sensitivity (see appendix B.2). In fact, in the multiparameter estimation schemes, $N$ different phases $\theta_k$ are encoded in the system via local generators (for us the individual generators $\hat s_{\vec{n},k}$). It is only in the particular case where all the phases are equal ($\theta_k=\theta,\:\forall k$) that we find the statistical factor $1/\sqrt{N}$ corresponding to the case of estimation of a single parameter $\theta$ encoded by the collective generator $\hat{S}_{\vec{n}}$.
2 Let us now come back to the case where the atoms are distributed in spatially separated modes. Unlike other multiparameter estimation schemes that require local measurements (i.e. measurements on individual atoms), our scheme requires only collective measurements. By "collective measurements" we mean measurements of the components of the collective spin operator (e.g. $\hat S_x$, $\hat S_y$ or $\hat S_z$), which indeed implies that the same spin component is measured for each atom. These collective measurements are naturally and usually performed in cold atoms experiments. What makes these measurements advantageous is the fact that it is not necessary to have the spatial resolution that would be required to perform measurements on individual atoms (i.e. local measurements).
Changes to the manuscript: we have added a footnote (footnote 1) to explain what we mean by a collective measurement, and a sentence in the end of the fourth paragraph of the introduction to explain its advantage with respect to local measurements.