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arXiv:1209.3547 · Physical Review A

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

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

Published Nov 13, 2012• 50 citations
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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 [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.

DOI
10.1103/physreva.86.053813

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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.
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.
The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cram\'er-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as ${\ensuremath{\omega}}^{\ensuremath{-}p}$ with p>1p>1p>1, the minimum mean-square error in any (single-time) phase estimate scales as ${\mathcal{N}}^{\ensuremath{-}2(p\ensuremath{-}1)/(p+1)}$, where N\mathcal{N}N is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit $p\ensuremath{\rightarrow}\ensuremath{\infty}$) and provides a stochastic Heisenberg limit for fluctuating phases. For p=2p=2p=2 (Brownian motion), this limit can be attained by phase tracking.
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.
Phase shift estimation with uncertainty below the Heisenberg limit, $\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}\ensuremath{\propto}1/{\overline{N}}_{T}$, where N‾T{\overline{N}}_{T}NT​ is the total average number of particles employed, is a mirage of linear quantum interferometry. Recently, Rivas and Luis, [New J. Phys. 14, 093052 (2012)] proposed a scheme to achieve a phase uncertainty $\ensuremath{\Delta}\ensuremath{\phi}\ensuremath{\propto}1/{\overline{N}}_{T}^{k}$, with kkk an arbitrary exponent. This sparked an intense debate in the literature which, ultimately, does not exclude the possibility to overcome $\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}$ at specific phase values. Our numerical analysis of the Rivas and Luis proposal shows that sub-Heisenberg uncertainties are obtained only when the estimator is strongly biased. No violation of the Heisenberg limit is found after bias correction or when using a bias-free Bayesian analysis.
There has been much interest in developing phase estimation schemes which beat the so-called Heisenberg limit, i.e., for which the phase resolution scales better than 1/n, where n is a measure of resources such as the average photon number or the number of atomic qubits. In particular, a number of nonlinear schemes have been proposed for which the resolution appears to scale as 1/n^k or even exp(-n), based on optimising the quantum Cramer-Rao bound. Such schemes include the use of entangled coherent states. However, it may be shown that the average root mean square errors of the proposed schemes (averaged over any prior distribution of phase shifts), cannot beat the Heisenberg limit, and that simple estimation schemes based on entangled coherent states cannot scale better than 1/n^{1/4}. This paradox is related to the role of 'bias' in Cramer-Rao bounds, and is only partially ameliorated via iterative implementations of the proposed schemes. The results are based on new information-theoretic bounds for the average information gain and error of any phase estimation scheme, and generalise to estimates of shifts generated by any operator having discrete eigenvalues.
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.