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

Dominic W. Berry, Mankei Tsang, Michael J W Hall, Howard M. Wiseman

Research output: Contribution to journalArticlepeer-review

56 Citations (Scopus)
25 Downloads (Pure)


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.

Original languageEnglish
Article number031018
Pages (from-to)1-28
Number of pages28
JournalPhysical Review X
Issue number3
Publication statusPublished - 18 Aug 2015

Bibliographical note

Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.


Dive into the research topics of 'Quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation'. Together they form a unique fingerprint.

Cite this