Maximum-likelihood statistics 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/Nₜₒₜ², where Nₜₒₜ is the mean number of photons available for all measurements. We have simulated their scheme for as many as 240 measurements and have found optimum phase sensitivities for 3{≤}Nₜₒₜ{≤}120; a power-law fit to the simulated data yields a phase sensitivity that scales as 1/Nₜₒₜ^0.82±0.01. By using a combination of numerical and analytical techniques, we extend our results to higher values of Nₜₒₜ than are accessible to our simulations; we find no evidence for phase sensitivities better than the benchmark 1/Nₜₒₜ sensitivity of squeezed-state interferometry. We conclude that reciprocal peak likelihood is not a good measure of phase sensitivity. We discuss other factors that are important to phase sensitivity.