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

## TL;DR

This paper compares weak-value amplification and optimal partitioning strategies for parameter estimation under correlated noise, showing that while WVA reduces variance, optimal analysis with partitioning offers slight improvements by utilizing correlations.

## Contribution

It provides an analytical and numerical comparison of WVA and optimal partitioning methods in correlated noise environments, highlighting the limitations and advantages of each approach.

## Key findings

- WVA reduces variance compared to conventional methods.
- Optimal partitioning slightly outperforms WVA by using correlations.
- Background subtraction is a special case of optimal partitioning.

## 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 noise model that can be analyzed exactly together with optimal statistical estimators, we compare WVA to a conventional measurement method. We find that introducing WVA indeed yields a much lower variance of the parameter of interest than does the conventional technique, 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 is because the simple WVA technique is not tailored to a given 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, in our model background subtraction turns out to be a special case of the optimal partitioning approach in the balanced case, possessing a similar typically slight advantage over WVA. These results give deeper insight into the role of partitioning measurements, with or without post-selection, in enhancing measurement precision, which some have found puzzling. We finish by presenting numerical results to model a more realistic laboratory situation of time-decaying correlations, showing our conclusions hold for a wide range of statistical models.