Quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation

We propose quantum versions of the Bell-Ziv-Zakai lower bounds for the error in multiparameter estimation. As an application, we consider measurement of a time-varying optical phase signal with stationary Gaussian prior statistics and a power-law spectrum ~1/|ω|^p, with p > 1. With no other assumptions, we show that the mean-square error has a lower bound scaling as 1/N²(^p-1)/^(p+1), where N is the time-averaged mean photon flux. Moreover, we show that this scaling is achievable by sampling and interpolation, for any p > 1. This bound is thus a rigorous generalization of the Heisenberg limit, for measurement of a single unknown optical phase, to a stochastically varying optical phase.