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arXiv:1703.01496 · Physical Review A

Weak-value amplification and optimal parameter estimation in the presence of correlated noise

Josiah Sinclair, Matin Hallaji, Aephraim M. Steinberg, Jeff Tollaksen, Andrew N. Jordan

Published Nov 27, 2017• 77 citations
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Abstract

We analytically and numerically investigate the performance of weak-value amplification (WVA) and related parameter estimation methods in the presence of temporally correlated noise. WVA is a special instance of a general measurement strategy that involves sorting data into separate subsets based on the outcome of a second ``partitioning'' measurement. Using a simplified correlated noise model that can be analyzed exactly together with optimal statistical estimators, we compare WVA to a conventional measurement method. We find that WVA indeed yields a much lower variance of the parameter of interest than the conventional technique does, optimized in the absence of any partitioning measurements. In contrast, a statistically optimal analysis that employs partitioning measurements, incorporating all partitioned results and their known correlations, is found to yield an improvement---typically slight---over the noise reduction achieved by WVA. This result occurs because the simple WVA technique is not tailored to any specific noise environment and therefore does not make use of correlations between the different partitions. We also compare WVA to traditional background subtraction, a familiar technique where measurement outcomes are partitioned to eliminate unknown offsets or errors in calibration. Surprisingly, for the cases we consider, background subtraction turns out to be a special case of the optimal partitioning approach, possessing a similar typically slight advantage over WVA. These results give deeper insight into the role of partitioning measurements (with or without postselection) in enhancing measurement precision, which some have found puzzling. They also resolve previously made conflicting claims about the usefulness of weak-value amplification to precision measurement in the presence of correlated noise. We finish by presenting numerical results to model a more realistic laboratory situation of time-decaying correlations, showing that our conclusions hold for a wide range of statistical models.

DOI
10.1103/physreva.96.052128

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We analytically and numerically investigate the performance of weak-value amplification (WVA) and related parameter estimation methods in the presence of temporally correlated noise. WVA is a special instance of a general measurement strategy that involves sorting data into separate subsets based on the outcome of a second ``partitioning'' measurement. Using a simplified correlated noise model that can be analyzed exactly together with optimal statistical estimators, we compare WVA to a conventional measurement method. We find that WVA indeed yields a much lower variance of the parameter of interest than the conventional technique does, optimized in the absence of any partitioning measurements. In contrast, a statistically optimal analysis that employs partitioning measurements, incorporating all partitioned results and their known correlations, is found to yield an improvement---typically slight---over the noise reduction achieved by WVA. This result occurs because the simple WVA technique is not tailored to any specific noise environment and therefore does not make use of correlations between the different partitions. We also compare WVA to traditional background subtraction, a familiar technique where measurement outcomes are partitioned to eliminate unknown offsets or errors in calibration. Surprisingly, for the cases we consider, background subtraction turns out to be a special case of the optimal partitioning approach, possessing a similar typically slight advantage over WVA. These results give deeper insight into the role of partitioning measurements (with or without postselection) in enhancing measurement precision, which some have found puzzling. They also resolve previously made conflicting claims about the usefulness of weak-value amplification to precision measurement in the presence of correlated noise. We finish by presenting numerical results to model a more realistic laboratory situation of time-decaying correlations, showing that our conclusions hold for a wide range of statistical models.
Weak value amplification (WVA) is a technique by which one can magnify the apparent strength of a measurement signal. Some have claimed that WVA can outperform more conventional measurement schemes in parameter estimation. Nonetheless, a significant body of theoretical work has challenged this perspective, suggesting WVA to be fundamentally suboptimal. Optimal measurements may not be practical, however. Two practical considerations that have been conjectured to afford a benefit to WVA over conventional measurement are certain types of noise and detector saturation. Here, we report a theoretical study of the role of saturation and pixel noise in WVA-based measurement, in which we carry out a Bayesian analysis of the Fisher information available using a saturable, pixelated, digitized, and/or noisy detector. We draw two conclusions: first, that saturation alone does not confer an advantage to the WVA approach over conventional measurement, and second, that WVA can outperform conventional measurement when saturation is combined with intrinsic pixel noise and/or digitization.
We show using statistically rigorous arguments that the technique of weak value amplification (WVA) does not perform better than standard statistical techniques for the tasks of single parameter estimation and signal detection. Specifically we prove that post-selection, a necessary ingredient for WVA, decreases estimation accuracy and, moreover, arranging for anomalously large weak values is a suboptimal strategy. In doing so, we explicitly provide the optimal estimator, which in turn allows us to identify the optimal experimental arrangement to be the one in which all outcomes have equal weak values (all as small as possible) and the initial state of the meter is the maximal eigenvalue of the square of the system observable. Finally, we give precise quantitative conditions for when weak measurement (measurements without post-selection or anomalously large weak values) can mitigate the effect of uncharacterized technical noise in estimation.
Recently, the weak-value amplification (WVA) has been extensively investigated as a promising quantum technique for improving an accuracy of a precision measurement.The basic idea of WVA comes from the weak measurement introduced by Aharonov, Albert, and Vaidman in 1988.The weak measurement is the indirect quantum measurement with the weak interaction and postselection of the final state of the measured system.The attention point is that we can measure the quantity called the weak value, which appears in the shift of the measuring probe given by the weak measurement.Because the weak value can be outside the range of the eigenvalues, the shift of the measuring probe obtained by the weak measurement becomes larger than the one given by the conventional measurement.This shift amplification effect by the weak value is the WVA, which has been studied for precisely measuring the small coupling constant.In this thesis, we propose the two possible applications of the WVA.First, we analytically obtain the optimal probe wave function for a given weak value by the variational method.It is shown that the amplification factor obtained by the optimal probe has noupper bound while the one in the Gaussian probe already reported by some researchers has the upper bound.Furthermore, the variance of the optimal probe after the measurement can be zero.We can derive the optimal probe by the Lagrange multiplier method for maximizing the shift of the probe position or minimizing the variance of the final distribution.Second we have developed the way which has a technical advantage of the WVA from the viewpoint of statistics.It is often argued that the WVA is not helpfull for the parameter estimation.Therefore we consider the statistical inferences that the WVA will be advantageous.We propose a method to determine whether the interaction is present or not and evaluate the capability of the WVA for this interaction detection problem by using the hypothesis testing in statistics.As the main result, it is shown that the merit of the WVA is the reduction of the possibility to miss the presence of the interaction more than the conventional measurement while keeping the probability of a misdetection, when the absolute value of the weak value is outside the range of the eigenvalues.In this discussion, we assume that the number of the obtained data is infinitely large and we neglect the data loss by the failure of the postseleciton.iii the deepest appreciation to my ex-supervisor
Weak-value amplification (WVA) has recently become an important technique for parameter estimation, owing to its ability to enhance the signal-to-noise ratio by amplifying extremely small signals with proper postselection strategies. In this paper, we propose an adaptive WVA scheme to achieve the highest Fisher information when using an unbalanced pointer. Different from previous schemes, the adaptive WVA scheme is associated with a real-time update on the postselection states with the help of feedback information from the outcomes, and the ``extremely small'' condition set on the parameter of interest is relaxed. By applying this scheme to a time-delay measurement scenario, we show by numerical simulation that the precision achieved in our scheme is several times higher than the standard WVA scheme. Our result might open a path for improving the WVA technique in a more flexible and robust way.
Abstract We present a theoretical investigation of weak-value amplification (WVA) under decoherence, quantifying its metrological capabilities through the quantum Fisher information (QFI). By modeling decoherence via Kraus operators acting before and after the weak measurement interaction, we derive exact expressions for the QFI governing parameter estimation of a weak coupling strength. These analytical results reveal the fundamental limitation imposed by decoherence on the QFI achievable via WVA. From these results, the optimal post-selection state that maximizes the QFI can be derived for different noise environments. Through paradigmatic examples, including amplitude damping and depolarizing channels, we demonstrate a key distinction: the optimal post-selection evolves with the noise strength in the amplitude damping channel, but is fixed in the depolarizing channel. This work provides both theoretical insights and practical guidance for optimizing metrological schemes based on WVA in realistic decoherent environments.
By choosing more orthogonality between preselection and postselection states, one can significantly improve the sensitivity in general optical quantum metrology based on the weak-value amplification (WVA) approach. However, increasing the orthogonality decreases the probability of detecting photons and makes the weak measurement difficult, especially when the weak measurement is disturbed by strong noise and the pointer is drowned in noise with a negative-decibel signal-to-noise ratio (SNR). In this article we introduce and numerically evaluate a modified weak-measurement protocol with a temporal pointer, namely, the autocorrelative weak-value amplification (AWVA) approach. Specifically, a small longitudinal time delay (tiny phase shift) $\ensuremath{\tau}$ of a Gaussian pulse is measured by implementing two simultaneous autocorrelative weak measurements under Gaussian white noise with different SNRs. The small quantities $\ensuremath{\tau}$ are obtained by measuring the autocorrelation coefficient of the pulses instead of fitting the shift of the mean value of the probe in the standard WVA technique. Simulation results show that the AWVA approach outperforms the standard WVA technique in the time domain with smaller statistical errors, remarkably increasing the precision of weak measurement under a strong noise background.