Fig. 7: FI for phase θ with imperfect measurements affected by the Poissonian noise.

a With binning strategies: the corresponding FI—\({\bar{F}}_{{{{{{{{\rm{2-bin}}}}}}}}}\) (orange) and \({\bar{F}}_{{{{{{{{\rm{3-bin}}}}}}}}}\) (blue) with optimal binning into two and three categories, respectively—compared against the exact \({\bar{{{{{{{{\mathcal{F}}}}}}}}}}^{({{{{{{{\rm{im}}}}}}}})}\) (ratio in %) computed by performing large enough cut-off (x≤100) in Eq. (9). The ratio of means for the Poissonnian distributions is set to \({\lambda }_{|1 \rangle }/{\lambda }_{|0 \rangle }=0.65\)⁷⁶, while \({\lambda }_{|0 \rangle }\) is varied. The inset shows the absolute values of FIs. b With binning strategies and the moment method: The FIs (F–black, F_2-bin–orange, F_3-bin–blue) presented now as a function of the input state angle ϕ = φ − θ (for \({\lambda }_{|1 \rangle }/{\lambda }_{|0 \rangle }=0.65\) and \({\lambda }_{\left|0\right\rangle }=27\)⁷⁶) in comparison to the lower bounds on F constructed by taking into account up to the second (F⁽¹⁾, light gray dash) and fourth moment (F⁽²⁾, dark gray dash) of the distribution describing the observed outcomes, \({q}_{\theta,\vec{\phi }}\). The vertical dotted lines indicate the (optimal) state angle at each of the respective quantities is maximised. Note that when the measurement is perfect, the FI is unity for all choices of the angle φ (not shown).