Quantum-enhanced multiparameter estimation and compressed sensing of a field

1-Framework 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

• The quantum gain due to correlations generated by OAT dynamics, $\xi(N,t)$, is the same in both strategies.
• The variance of the estimated parameters $\Delta\theta_k$ is not the same where the "Scanning Microscope" strategy has an advantage of a statistical factor $1/\sqrt N$. This comes from the fact that each parameter is encoded on an ensemble of $N$ atoms in the "Scanning Microscope" strategy while, in our strategy, each parameter is encoded on only a single atom.
• In the strategy we consider all the atoms remain in fixed positions and do not interact while the "Scanning Microscope" strategy requires physically moving all the atoms constituting the sensor, which exposes it to problems related to the interaction between atoms: "over squeezing" and atom loss, and, if one wants to keep the ensemble diluted, it reduces the spatial resolution of the sensor.
• Another advantage of our strategy is that it naturally allows for "compressed sensing" by measuring only the first $L_{\cal H}<N$ Hadamard coefficients of the discretized field on the lattice, as we show in figure 5. This, in contrast, is not allowed in the "Scanning Microscope" strategy.

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.