Quantum Bell-Ziv-Zakai Bounds and Heisenberg Limits for Waveform Estimation
Dominic W. Berry, Mankei Tsang, Michael J. W. Hall, Howard M. Wiseman · Physical Review X · 2015
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=jωj p , with p > 1.With no other assumptions, we show that the mean-square error has a lower bound scaling as 1=N 2ðp-1Þ=ðpþ1Þ , 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.