Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates

Dominic W. Berry, Michael J W Hall, Marcin Zwierz, Howard M. Wiseman

Research output: Contribution to journalArticleResearchpeer-review

Abstract

The ultimate bound to the accuracy of phase estimates is often assumed to be given by the Heisenberg limit. Recent work seemed to indicate that this bound can be violated, yielding measurements with much higher accuracy than was previously expected. The Heisenberg limit can be restored as a rigorous bound to the accuracy provided one considers the accuracy averaged over the possible values of the unknown phase, as we have recently shown. Here we present an expanded proof of this result together with a number of additional results, including the proof of a previously conjectured stronger bound in the asymptotic limit. Other measures of the accuracy are examined, as well as other restrictions on the generator of the phase shifts. We provide expanded numerical results for the minimum error and asymptotic expansions. The significance of the results claiming violation of the Heisenberg limit is assessed, followed by a detailed discussion of the limitations of the Cramér-Rao bound.

LanguageEnglish
Article number053813
Pages1-20
Number of pages20
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Volume86
Issue number5
DOIs
Publication statusPublished - 13 Nov 2012

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estimates
Cramer-Rao bounds
constrictions
phase shift
generators
expansion

Bibliographical note

Berry, DW, Hall, MJW, Zwierz, M. & Wiseman, HM. Physical review A. Atomic, molecular, and optical physics, 86(5), 053813, 2012. Copyright 2012 by the American Physical Society. The original article can be found at http://dx.doi.org/10.1103/PhysRevA.86.053813.

Cite this

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abstract = "The ultimate bound to the accuracy of phase estimates is often assumed to be given by the Heisenberg limit. Recent work seemed to indicate that this bound can be violated, yielding measurements with much higher accuracy than was previously expected. The Heisenberg limit can be restored as a rigorous bound to the accuracy provided one considers the accuracy averaged over the possible values of the unknown phase, as we have recently shown. Here we present an expanded proof of this result together with a number of additional results, including the proof of a previously conjectured stronger bound in the asymptotic limit. Other measures of the accuracy are examined, as well as other restrictions on the generator of the phase shifts. We provide expanded numerical results for the minimum error and asymptotic expansions. The significance of the results claiming violation of the Heisenberg limit is assessed, followed by a detailed discussion of the limitations of the Cram{\'e}r-Rao bound.",
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Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates. / Berry, Dominic W.; Hall, Michael J W; Zwierz, Marcin; Wiseman, Howard M.

In: Physical Review A - Atomic, Molecular, and Optical Physics, Vol. 86, No. 5, 053813, 13.11.2012, p. 1-20.

Research output: Contribution to journalArticleResearchpeer-review

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