Photon-limited bounds for phase retrieval

We show that the optimal Cramér-Rao lower bound on the mean-square error for the estimation of a coherent signal from photon-limited intensity measurements is equal to the number of signal elements, or the number of signal elements minus one when we account for the unobservable reference phase. Whereas this bound is attained by phase-quadrature holography, we also show that it can be attained through a phase-retrieval system that does not require a coherent reference. We also present the bounds for classic phase-retrieval and ptychography, and show that practical coding strategies can approach optimal performance.