Optimal observables for quantum measurements with noisy squeezed spin states

Certain multipartite entangled states enable measurements with sensitivities that surpass the shot-noise limit. Among them, squeezed spin states form an important class and achieve sub-shot-noise sensitivity in Ramsey interferometry. In spin-squeezing experiments in solid-state systems, it is challenging to reduce the squeezing parameter below the standard quantum limit (SQL), primarily due to inhomogeneous broadening of spins. Nevertheless, numerical simulations suggest that the Cramér–Rao lower bound for such noisy squeezed spin states can still surpass the SQL, indicating that these states could be utilized to achieve sub-shot-noise sensitivity in quantum measurements.

In our presentation, we numerically calculate noisy squeezed spin states that maximize the quantum Fisher information for static magnetic field measurements and determine the sets of observables that maximize the classical Fisher information. The optimal observables are assumed to be expandable in terms of magnetic multipole moments up to the quadrupole order, with their coefficients obtained using a Bayesian optimization method. The numerical results suggest that measurements employing noisy squeezed spin states can achieve sub-shot-noise sensitivity by measuring appropriate combinations of magnetic multipole moments. Results including higher-order multipoles will also be discussed.