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 journalArticleResearchpeer-review

Abstract

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.

LanguageEnglish
Article number031018
Pages1-28
Number of pages28
JournalPhysical Review X
Volume5
Issue number3
DOIs
Publication statusPublished - 2015

Fingerprint

bells
waveforms
scaling
interpolation
sampling
statistics
photons

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.

Cite this

Berry, Dominic W. ; Tsang, Mankei ; Hall, Michael J W ; Wiseman, Howard M. / Quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation. In: Physical Review X. 2015 ; Vol. 5, No. 3. pp. 1-28.
@article{f417f0aa5ace4ef5b44ca75220a7098f,
title = "Quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation",
abstract = "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.",
author = "Berry, {Dominic W.} and Mankei Tsang and Hall, {Michael J W} and Wiseman, {Howard M.}",
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.",
year = "2015",
doi = "10.1103/PhysRevX.5.031018",
language = "English",
volume = "5",
pages = "1--28",
journal = "Physical Review X",
issn = "2160-3308",
publisher = "American Physical Society",
number = "3",

}

Quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation. / Berry, Dominic W.; Tsang, Mankei; Hall, Michael J W; Wiseman, Howard M.

In: Physical Review X, Vol. 5, No. 3, 031018, 2015, p. 1-28.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - Quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation

AU - Berry, Dominic W.

AU - Tsang, Mankei

AU - Hall, Michael J W

AU - Wiseman, Howard M.

N1 - 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.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84946027791&partnerID=8YFLogxK

U2 - 10.1103/PhysRevX.5.031018

DO - 10.1103/PhysRevX.5.031018

M3 - Article

VL - 5

SP - 1

EP - 28

JO - Physical Review X

T2 - Physical Review X

JF - Physical Review X

SN - 2160-3308

IS - 3

M1 - 031018

ER -