Maximum-likelihood statistics of multiple quantum phase measurements

Abstract

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 available for all measurements. We have simulated their scheme for as many as 240 measurements and have found optimum phase sensitivities for 3\ensuremath{\le}${\mathit{N}}_{\mathrm{tot}}$\ensuremath{\le}120; a power-law fit to the simulated data yields a phase sensitivity that scales as 1/${\mathit{N}}_{\mathrm{tot}}^{0.82\ifmmode\pm\else\textpm\fi{}0.01}$. By using a combination of numerical and analytical techniques, we extend our results to higher values of ${\mathit{N}}_{\mathrm{tot}}$ than are accessible to our simulations; we find no evidence for phase sensitivities better than the benchmark 1/${\mathit{N}}_{\mathrm{tot}}$ 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.