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arXiv:1105.6310 · New Journal of Physics

Sub-Heisenberg estimation of non-random phase shifts

Ángel Rivas, Alfredo Luis

Published Sep 26, 2012• 58 citations
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Abstract

We provide evidence that the uncertainty in detection of small and deterministic phase-shift deviations from a working point can be lower than the Heisenberg bound, for a fixed finite mean number of photons. We achieve that by exploiting nonlinearity of estimators and coherence with the vacuum.
DOI
10.1088/1367-2630/14/9/093052

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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.
We discuss an adaptive interferometric approach to quantum estimation of phase shifts. In a first step we find optimized two-mode input states for a generalized Mach-Zehnder interferometer. These two states show a small phase variance at the so-called Heisenberg limit. It turns out that they can be generated by entangling a coherent state and a squeezed vacuum at a beam splitter. Moreover, the interferometric setup allows us to describe an adaptive estimation scheme for arbitrary phase shifts if we assume a fixed mean photon number as a quantum resource.
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.
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.