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 similar to 1/vertical bar omega vertical bar(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.