Quantum-enhanced multiparameter estimation and compressed sensing of a field

We show that a significant quantum gain corresponding to squeezed or over-squeezed spin states can be obtained in multiparameter estimation by measuring the Hadamard coefficients of 1D 2D signal. The physical platform we consider consists two-level atoms an optical lattice squeezed-Mott configuration, more generally correlated spins distributed spatially separated modes. Our protocol requires possibility locally flip spins, but relies on collective measurements. give examples applications scalar vector field mapping and compressed sensing.