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 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=jωjp, with p > 1. With no other assumptions, we show that the mean-square error has a lower bound scaling as 1=N2ðp−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.
Physical Review X
© The Author(s) 2015. Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License (http://creativecommons.org/licenses/by/3.0/) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, providing that the work is properly cited.
Quantum Information, Computation and Communication