Stochastic heisenberg limit

optimal estimation of a fluctuating phase

Dominic W. Berry*, Michael J W Hall, Howard M. Wiseman

*Corresponding author for this work

Research output: Contribution to journalArticle

23 Citations (Scopus)
6 Downloads (Pure)


The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cramér-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as ω-p with p>1, the minimum mean-square error in any (single-time) phase estimate scales as N -2(p-1)/(p+1), where N is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit p→∞) and provides a stochastic Heisenberg limit for fluctuating phases. For p=2 (Brownian motion), this limit can be attained by phase tracking.

Original languageEnglish
Article number113601
Pages (from-to)1-5
Number of pages5
JournalPhysical Review Letters
Issue number11
Publication statusPublished - 13 Sep 2013

Bibliographical note

Berry, D. W., Hall, M. J. W., & Wiseman, H. M. (2013). Stochastic heisenberg limit : optimal estimation of a fluctuating phase. Physical review letters, 111(11), 113601, 2013. Copyright 2013 by the American Physical Society. The original article can be found at

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