Maximum-likelihood analysis of multiple quantum phase measurements

Shapiro, Shepard, and Wong [Phys. Rev. Lett. 62, 2377 (1989)] suggested that a scheme of multiple phase measurements, using quantum states with minimum ‘‘reciprocal peak likelihood,’’ could achieve a phase sensitivity scaling as 1/${\mathit{N}}_{\mathrm{tot}}^2$, where ${\mathit{N}}_{\mathrm{tot}}$ is the mean number of photons availavle for all measurements. We have simulated their scheme for as many as 240 measurements and have found optimum phase sensitivities for 3≤${\mathit{N}}_{\mathrm{tot}}$≤120. A power-law fit to the simulated data yields a phase sensitivity that scales as 1/${\mathit{N}}_{\mathrm{tot}}^{0.85\pm 0.01}$. We conclude that reciprocal peak likelihood is not a good measure of sensitivity.