Fig. 6: Attaining the optimal SS of precision in presence of measurement imperfections and local unitary control.

The thick solid black line depicts the MSE in estimating phase θ from an imperfect measurement of the angular momentum operator \({\hat{J}}_{{{{{{{{\rm{x}}}}}}}}}\), while the N qubit probes are prepared in a one-axis spin-squeezed state⁵⁷, optimised by local control (see Methods). The noisy detection channel corresponds to an asymmetric bit-flip map with probabilities \({\mathsf{p}}=0.95\) and \({\mathsf{q}}=0.9\). The dotted black line denotes the asymptotic CE bound with \({\bar{F}}_{N}^{({{{{{{{\rm{CE}}}}}}}},{{{{{{{\rm{as}}}}}}}})}\) given by Eq. (24), while the thin red solid line is the exact achievable precision \(1/{\bar{{{{{{{{\mathcal{F}}}}}}}}}}_{N}^{({{{{{{{\rm{im}}}}}}}},{{{{{{{\rm{l}}}}}}}})}\), which we compute numerically up to N = 6 by brute-force heuristic methods. The dashed black line corresponds to \(1/{F}_{N}^{{{{{{{{\rm{(CE)}}}}}}}}}\) in Eq. (21) applicable in absence of control (\({\forall }_{\ell }:\ {V}_{{\vec{\phi }}_{\ell }}^{(\ell )}={\mathbb{1}}\)). At small N (≲4), the ultimate precision can be attained by performing (imperfect) parity measurements with input GHZ states (thin blue line). For comparison, we also include the optimal precision attained by uncorrelated probe states, \(1/(N{\bar{{{{{{{{\mathcal{F}}}}}}}}}}_{1}^{({{{{{{{\rm{im}}}}}}}})})\), (solid gray).