The quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation

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We propose quantum versions of the Bell-Ziv-Zakai lower bounds on 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 accuracy 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.