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arXiv:1904.00557 · Optics Express

Multi-outcome homodyne detection in a coherent-state light interferometer

J. Z. Wang, Z. Q. Yang, A. X. Chen, W. Yang, G. R. Jin

Published Mar 29, 2019• 8 citations
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Abstract

The Cramér-Rao bound plays a central role in both classical and quantum parameter estimation, but finding the observable and the resulting inversion estimator that saturates this bound remains an open issue for general multi-outcome measurements. Here we consider multi-outcome homodyne detection in a coherent-light Mach-Zehnder interferometer and construct a family of inversion estimators that almost saturate the Cramér-Rao bound over the whole range of phase interval. This provides a clue on constructing optimal inversion estimators for phase estimation and other parameter estimation in any multi-outcome measurement.

DOI
10.1364/oe.27.010343

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The Cramér-Rao bound plays a central role in both classical and quantum parameter estimation, but finding the observable and the resulting inversion estimator that saturates this bound remains an open issue for general multi-outcome measurements. Here we consider multi-outcome homodyne detection in a coherent-light Mach-Zehnder interferometer and construct a family of inversion estimators that almost saturate the Cramér-Rao bound over the whole range of phase interval. This provides a clue on constructing optimal inversion estimators for phase estimation and other parameter estimation in any multi-outcome measurement.
As the simplest phase estimation protocol, the inversion estimator has been widely adopted in the quantum metrology, since its performance can be simply quantified by the error-propagation formula. For a general binary-outcome measurement, we show that both the inversion estimator and the maximum likelihood estimator are the same and can asymptotically saturate the Cramér-Rao lower bound. These observations are applied to the parity detection and the zero-nonzero photon counting that have achieved super-resolved phase measurement near the shot-noise phase sensitivity in a coherent-state light Mach–Zehnder interferometer. We numerically show that the phase uncertainty almost follows the theoretical prediction of the lower bound. At the optimal working point, the best sensitivities also reach their associated analytic results for the two binary-outcome measurement schemes.
Due to the potential of quantum advantage to surpass the standard quantum limit (SQL), the nonlinear interferometers have garnered significant attention from researchers in the field of precision measurement. However, many practical applications require multi-parameter estimation. In this work, we discuss the precision limit of multi-parameter estimation of pure Gaussian states based on nonlinear interferometers, and derive the Holevo Cramér-Rao Bound (HCRB) for the case where both modes undergo displacement estimation. Furthermore, we compare our analytical results with the quantum Cramér-Rao Bound based on the symmetric logarithmic derivative (SLD-CRB), and with the result of the dual homodyne measurement. Through numerical analysis, we find that the HCRB equals the result of the dual homodyne measurement, whereas SLD-CRB is not saturable at small squeezed parameters. Therefore, this indicates that the HCRB is tight. Additionally, we provide intuitive analysis and visual representation of our numerical results in phase space.
Finding the optimal estimator capable of saturating the Cram\'er-Rao bound is important for quantum parameter estimation. The maximum likelihood estimator (MLE) is well known to be asymptotically optimal, but it is usually nonanalytic and hence requires extensive numerical computation. On the other hand, the inversion estimator is widely used in experiments due to its simplicity, but it is usually suboptimal. Here we derive an estimator based on the linear combination of multiple inversion estimators associated with the occurrence probability of different measurement outcomes. This linear combination estimator shares the asymptotic optimality of the MLE and the simplicity of the inversion estimator and is applicable to general multi-outcome measurements, irrespective of the specific system and the noise in the system. We demonstrate this estimator for the intensity-difference measurement to a six-photon twin-Fock input state.
Optimal measurement scheme with an efficient data processing is important in quantum-enhanced interferometry. Here we prove that for a general binary-outcome measurement, the simplest data processing based on inverting the average signal can saturate the Cram\'er-Rao bound. This idea is illustrated by binary-outcome homodyne detection, even-odd photon counting (i.e., parity detection), and zero-nonzero photon counting that have achieved super-resolved interferometric fringe and shot-noise limited sensitivity in coherent-light Mach-Zehnder interferometer. The roles of phase diffusion are investigated in these binary-outcome measurements. We find that the diffusion degrades the fringe resolution and the achievable phase sensitivity. Our analytical results confirm that the zero-nonzero counting can produce a slightly better sensitivity than that of the parity detection, as demonstrated in a recent experiment.
Performing homodyne detection at one port of a squeezed-state light interferometer and then binarizing the measurement data are important to achieve superresolving and supersensitive phase measurements. Here we propose an alternative data-processing technique by dividing the measurement quadrature into three bins (equivalent to a multi-outcome measurement), which leads to a higher improvement in the phase resolution and the phase sensitivity under realistic experimental conditions. Furthermore, we develop an alternative phase-estimation protocol based on a combination of the inversion estimators of each outcome and show that the estimator can saturate the Cram\'er-Rao lower bound, similar to the asymptotically unbiased maximum likelihood estimator.
Due to the potential of quantum advantage to surpass the standard quantum limit (SQL), nonlinear interferometers have garnered significant attention from researchers in the field of precision measurement. However, many practical applications require multiparameter estimation. In this work, we discuss the precision limit of multi-parameter estimation of pure Gaussian states based on nonlinear interferometers, and derive the Holevo Cramér−Rao bound (HCRB) for the case where both modes undergo displacement estimation. Furthermore, we compare our analytical results with the quantum Cramér−Rao bound based on the symmetric logarithmic derivative (SLD-CRB), and with the result of the dual homodyne measurement. Through numerical analysis, we find that the HCRB equals the result of the dual homodyne measurement, whereas SLD-CRB is not saturable at small squeezed parameters. Therefore, this indicates that the HCRB is tight. Additionally, we provide intuitive analysis and visual representation of our numerical results in phase space.