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arXiv:1403.0955

Upper bounds on the quantum Fisher Information in the presence of general dephasing

Katarzyna Macieszczak

Published Mar 4, 2014• 3 citations
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Abstract

We derive upper bounds on the quantum Fisher information in interferometry with $N$ subsystems, e.g. two-level atoms or Gaussian modes, in the presence of arbitrarily correlated Gaussian dephasing including independent and collective dephasing. The derived upper bound enables us to analyse the Fisher information asymptotic behaviour when $N\rightarrow\infty$. Dephasing introduces random phases to subsystems dynamics, which lowers the precision of estimating the phase difference $ϕ$ in an interferometer. The method presented uses Bayesian estimation of the random phases and eliminates dephasing noise by calculating their weighted arithmetic mean, which correponds to the phase $ϕ$ estimated in interferometry.

DOI
10.48550/arxiv.1403.0955

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We advocate a Bayesian approach to optimal quantum frequency estimation—an important issue for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of prior knowledge in determining the optimal interrogation times and optimal estimation strategies. We propose a general framework capable of describing local oscillator noise as well as additional collective atomic dephasing effects. For a Gaussian noise, the average Bayesian cost can be expressed using the quantum Fisher information. Thus we establish a direct link between the two, often competing, approaches to quantum estimation theory.
We advocate a Bayesian approach to optimal quantum frequency estimation—an important issue for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of prior knowledge in determining the optimal interrogation times and optimal estimation strategies. We propose a general framework capable of describing local oscillator noise as well as additional collective atomic dephasing effects. For a Gaussian noise, the average Bayesian cost can be expressed using the quantum Fisher information. Thus we establish a direct link between the two, often competing, approaches to quantum estimation theory.
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The Fisher information FFF gives a limit to the ultimate precision achievable in a phase estimation protocol. It has been shown recently that the Fisher information for a linear two-mode interferometer cannot exceed the number of particles if the input state is separable. As a direct consequence, with such input states the shot-noise limit is the ultimate limit of precision. In this work, we go a step further by deducing bounds on FFF for several multiparticle entanglement classes. These bounds imply that genuine multiparticle entanglement is needed for reaching the highest sensitivities in quantum interferometry. We further compute similar bounds on the average Fisher information F‾\overline{F}F for collective spin operators, where the average is performed over all possible spin directions. We show that these criteria detect different sets of states and illustrate their strengths by considering several examples, also using experimental data. In particular, the criterion based on F‾\overline{F}F is able to detect certain bound entangled states.
A major obstacle to attaining the fundamental precision limit of phase estimation in interferometry is the identification and implementation of the optimal measurement. Here we demonstrate that this can be accomplished by the use of three conventional measurements among interferometers with Bayesian estimation techniques. Conditions that hold for the precision limit to be attained with these measurements are obtained by explicitly calculating the Fisher information. Fortunately, these conditions are naturally satisfied in most interferometric experiments. We apply our results to an atomic spectroscopy experiment and examine the robustness of phase sensitivity for the two-axis counter-twisted state suffering from detection noise.
Dephasing noise is a ubiquitous source of decoherence in current atomic sensors. We address the problem of entanglement-assisted frequency estimation subject to classical dephasing noise with full spatial correlations (collective) and arbitrary temporal correlations. Our contributions are threefold. (i) We derive rigorous, state-independent bounds on the achievable estimation precision, showing how they are entirely determined by the short-time behavior of the decoherence function. For temporally uncorrelated (Markovian) dephasing, precision is limited by a probe-independent constant. For temporally correlated stationary noise, the bound approaches the noiseless limit for classical states, precluding any asymptotic quantum advantage. (ii) We show that these scaling bounds are tight, by constructing generalized Ramsey protocols that saturate them. These optimal protocols use squeezing at the input and before readout, both of which are available in state-of-the-art atomic interferometers. Implementing a perfect-echo protocol, which reaches Heisenberg scaling in the absence of noise, remains optimal in this noisy setting, irrespective of the noise temporal correlations. (iii) We prove that arbitrary collective open-loop control cannot lift the no-go for super-classical precision scaling under either Markovian or colored stationary noise, highlighting the detrimental nature of full spatial correlations. In the latter case, temporal correlations may nonetheless enable constant-factor improvements over the standard quantum limit, which may still be important in practical metrological scenarios.