Fig. 5: Attaining the ultimate HS of precision in presence of measurement imperfections and global unitary control.

The case of phase estimation with N qubit probes is considered, which are initialised in a GHZ state, whereas the outcomes of ideal local measurements undergo an asymmetric bit-flip channel with \({\mathsf{p}}\) = 0.95 and \({\mathsf{q}}\) = 0.9. The black solid line is the exact (numerical) FI for a specific choice of the control global unitary \({V}_{\vec{\Phi }}\), while the black dotted line is its lower bound \({F}_{N}^{\downarrow }({V}_{\vec{\Phi }})\) defined in Eq. (20)—both converge to the optimal achievable \({{{{{{{\mathcal{F}}}}}}}}[{\psi }^{N}(\theta )]={N}^{2}\) (black dashed line). The family of lines in blue are the corresponding FIs for the case of a distorted GHZ state, with an admixture of white noise (with r = 0.7 in Eq. (18)) being added.