Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1201.4542 · New Journal of Physics

Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information

Michael J. W. Hall, Howard M. Wiseman

Published Mar 28, 2012• 69 citations
arXivHTMLPDF

Abstract

A rigorous lower bound is obtained for the average resolution of any estimate of a shift parameter, such as an optical phase shift or a spatial translation. The bound has the asymptotic form k I /〈2| G |〉 where G is the generator of the shift (with an arbitrary discrete or continuous spectrum), and hence establishes a universally applicable bound of the same form as the usual Heisenberg limit. The scaling constant k I depends on prior information about the shift parameter. For example, in phase sensing regimes, where the phase shift is confined to some small interval of length L , the relative resolution has the strict lower bound (2 π e 3 ) −1/2 /〈2 m | G 1 | + 1〉, where m is the number of probes, each with generator G 1 , and entangling joint measurements are permitted. Generalizations using other resource measures and including noise are briefly discussed. The results rely on the derivation of general entropic uncertainty relations for continuous observables, which are of interest in their own right.

DOI
10.1088/1367-2630/14/3/033040

Related papers

Universality of the Heisenberg limit for estimates of random phase shifts

Michael J. W. Hall, Dominic William Berry, Marcin Zwierz +1 more

The Heisenberg limit traditionally provides a lower bound on the phase uncertainty scaling as $1/\ensuremath{\langle}N\ensuremath{\rangle}$, where $\ensuremath{\langle}N\ensuremath{\rangle}$ is the mean number of photons in the probe. However, this limit has a number of loopholes which potentially might be exploited to achieve measurements with even greater accuracy. Here we close these loopholes by proving a completely rigorous form of the Heisenberg limit for the average error over all phase shifts, applicable to any estimate of a completely unknown phase shift. Our result gives a completely general, constraint-free, and nonasymptotic statement of the Heisenberg limit. It holds for all phase estimation schemes, including multiple passes, nonlinear phase shifts, multimode probes, and arbitrary measurements.

arXiv:1111.0788•Physical Review A•Apr 13, 2012•78 citations

Better Heisenberg Limits, Coherence Bounds, and Energy-Time Tradeoffs via Quantum Rényi Information

Michael J. W. Hall

An uncertainty relation for the Rényi entropies of conjugate quantum observables is used to obtain a strong Heisenberg limit of the form RMSE≥f(α)/(⟨N⟩+12), bounding the root mean square error of any estimate of a random optical phase shift in terms of average photon number, where f(α) is maximised for non-Shannon entropies. Related simple yet strong uncertainty relations linking phase uncertainty to the photon number distribution, such as ΔΦ≥maxnpn, are also obtained. These results are significantly strengthened via upper and lower bounds on the Rényi mutual information of quantum communication channels, related to asymmetry and convolution, and applied to the estimation (with prior information) of unitary shift parameters such as rotation angle and time, and to obtain strong bounds on measures of coherence. Sharper Rényi entropic uncertainty relations are also obtained, including time-energy uncertainty relations for Hamiltonians with discrete spectra. In the latter case almost-periodic Rényi entropies are introduced for nonperiodic systems.

arXiv:2210.14613•Entropy•Nov 17, 2022•2 citations

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

Dominic William Berry, Michael J. W. Hall, Marcin Zwierz +1 more

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 [Phys. Rev. A 85, 041802(R) (2012)]. 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\'er-Rao bound.

arXiv:1209.3547•Physical Review A•Nov 13, 2012•50 citations

Entropic Heisenberg limits and uncertainty relations from the Holevo information bound

Michael J. W. Hall

Abstract Strong and general entropic and geometric Heisenberg limits are obtained, for estimates of multiparameter unitary displacements in quantum metrology, such as the estimation of a magnetic field from the induced rotation of a probe state in three dimensions. A key ingredient is the Holevo bound on the Shannon mutual information of a quantum communication channel. This leads to a Bayesian bound on performance, in terms of the prior distribution of the displacement and the asymmetry of the input probe state with respect to the displacement group. A geometric measure of performance related to entropy is proposed for general parameter estimation. It is also shown how strong entropic uncertainty relations for mutually unbiased observables, such as number and phase, position and momentum, energy and time, and orthogonal spin-1/2 directions, can be obtained from elementary applications of Holevo’s bound. A geometric interpretation of results is emphasised, in terms of the ‘volumes’ of quantum and classical statistical ensembles.

arXiv:1804.01343•Journal of Physics A Mathematical and Theoretical•Jul 23, 2018•13 citations

Sub-Heisenberg Estimation Strategies Are Ineffective

Vittorio Giovannetti, Lorenzo Maccone

In interferometry, sub-Heisenberg strategies claim to achieve a phase estimation error smaller than the inverse of the mean number of photons employed (Heisenberg bound). Here we show that one can achieve a comparable precision without performing any measurement, just using the large prior information that sub-Heisenberg strategies require. For uniform prior (i.e., no prior information), we prove that these strategies cannot achieve more than a fixed gain of about 1.73 over Heisenberg-limited interferometry. Analogous results hold for arbitrary single-mode prior distributions. These results extend also beyond interferometry: the effective error in estimating any parameter is lower bounded by a quantity proportional to the inverse expectation value (above a ground state) of the generator of translations of the parameter.

arXiv:1201.1878•Physical Review Letters•May 22, 2012•90 citations

Quantum Bell-Ziv-Zakai Bounds and Heisenberg Limits for Waveform Estimation

Dominic William Berry, Mankei Tsang, Michael J. W. Hall +1 more

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=jωj p , with p > 1.With no other assumptions, we show that the mean-square error has a lower bound scaling as 1=N 2ðp-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.

arXiv:1409.7877•Physical Review X•Aug 18, 2015•81 citations
A rigorous lower bound is obtained for the average resolution of any estimate of a shift parameter, such as an optical phase shift or a spatial translation. The bound has the asymptotic form k I /〈2| G |〉 where G is the generator of the shift (with an arbitrary discrete or continuous spectrum), and hence establishes a universally applicable bound of the same form as the usual Heisenberg limit. The scaling constant k I depends on prior information about the shift parameter. For example, in phase sensing regimes, where the phase shift is confined to some small interval of length L , the relative resolution has the strict lower bound (2 π e 3 ) −1/2 /〈2 m | G 1 | + 1〉, where m is the number of probes, each with generator G 1 , and entangling joint measurements are permitted. Generalizations using other resource measures and including noise are briefly discussed. The results rely on the derivation of general entropic uncertainty relations for continuous observables, which are of interest in their own right.
The Heisenberg limit traditionally provides a lower bound on the phase uncertainty scaling as $1/\ensuremath{\langle}N\ensuremath{\rangle}$, where $\ensuremath{\langle}N\ensuremath{\rangle}$ is the mean number of photons in the probe. However, this limit has a number of loopholes which potentially might be exploited to achieve measurements with even greater accuracy. Here we close these loopholes by proving a completely rigorous form of the Heisenberg limit for the average error over all phase shifts, applicable to any estimate of a completely unknown phase shift. Our result gives a completely general, constraint-free, and nonasymptotic statement of the Heisenberg limit. It holds for all phase estimation schemes, including multiple passes, nonlinear phase shifts, multimode probes, and arbitrary measurements.
An uncertainty relation for the Rényi entropies of conjugate quantum observables is used to obtain a strong Heisenberg limit of the form RMSE≥f(α)/(⟨N⟩+12), bounding the root mean square error of any estimate of a random optical phase shift in terms of average photon number, where f(α) is maximised for non-Shannon entropies. Related simple yet strong uncertainty relations linking phase uncertainty to the photon number distribution, such as ΔΦ≥maxnpn, are also obtained. These results are significantly strengthened via upper and lower bounds on the Rényi mutual information of quantum communication channels, related to asymmetry and convolution, and applied to the estimation (with prior information) of unitary shift parameters such as rotation angle and time, and to obtain strong bounds on measures of coherence. Sharper Rényi entropic uncertainty relations are also obtained, including time-energy uncertainty relations for Hamiltonians with discrete spectra. In the latter case almost-periodic Rényi entropies are introduced for nonperiodic systems.
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 [Phys. Rev. A 85, 041802(R) (2012)]. 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\'er-Rao bound.
Abstract Strong and general entropic and geometric Heisenberg limits are obtained, for estimates of multiparameter unitary displacements in quantum metrology, such as the estimation of a magnetic field from the induced rotation of a probe state in three dimensions. A key ingredient is the Holevo bound on the Shannon mutual information of a quantum communication channel. This leads to a Bayesian bound on performance, in terms of the prior distribution of the displacement and the asymmetry of the input probe state with respect to the displacement group. A geometric measure of performance related to entropy is proposed for general parameter estimation. It is also shown how strong entropic uncertainty relations for mutually unbiased observables, such as number and phase, position and momentum, energy and time, and orthogonal spin-1/2 directions, can be obtained from elementary applications of Holevo’s bound. A geometric interpretation of results is emphasised, in terms of the ‘volumes’ of quantum and classical statistical ensembles.
In interferometry, sub-Heisenberg strategies claim to achieve a phase estimation error smaller than the inverse of the mean number of photons employed (Heisenberg bound). Here we show that one can achieve a comparable precision without performing any measurement, just using the large prior information that sub-Heisenberg strategies require. For uniform prior (i.e., no prior information), we prove that these strategies cannot achieve more than a fixed gain of about 1.73 over Heisenberg-limited interferometry. Analogous results hold for arbitrary single-mode prior distributions. These results extend also beyond interferometry: the effective error in estimating any parameter is lower bounded by a quantity proportional to the inverse expectation value (above a ground state) of the generator of translations of the parameter.
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=jωj p , with p > 1.With no other assumptions, we show that the mean-square error has a lower bound scaling as 1=N 2ðp-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.