Variance Control in Weak Value Measurement Pointers

TL;DR

This paper analyzes how to control the variance of pointer observables in weak value measurements, revealing conditions under which measurement can reduce variance and improve measurement sensitivity.

Contribution

It derives expressions for pointer variance in weak measurements with complex weak values, identifying conditions for variance reduction and enhanced measurement sensitivity.

Findings

Variance expressions include terms linked to the weak value's imaginary part.

Control conditions can reduce pointer variance post-measurement.

Potential for improved measurement accuracy through variance control.

Abstract

The variance of an arbitrary pointer observable is considered for the general case that a complex weak value is measured using a complex valued pointer state. For the typical cases where the pointer observable is either its position or momentum, the associated expressions for the pointer's variance after the measurement contain a term proportional to the product of the weak value's imaginary part with the rate of change of the third central moment of position relative to the initial pointer state just prior to the time of the measurement interaction when position is the observable - or with the initial pointer state's third central moment of momentum when momentum is the observable. These terms provide a means for controlling pointer position and momentum variance and identify control conditions which - when satisfied - can yield variances that are smaller after the measurement than they were before the measurement. Measurement sensitivities which are useful for estimating weak value measurement accuracies are also briefly discussed.