Full counting statistics of weak measurement

TL;DR

This paper develops a comprehensive framework for analyzing the full counting statistics of weak measurements, introducing characteristic functions and normal weak values to describe measurement outcomes beyond averages and variances.

Contribution

It provides the first complete characterization of weak measurement statistics using characteristic functions and introduces normal weak values for a unified description.

Findings

The characteristic function for weak measurement moments is derived.

Weak measurement statistics can be described by a complex weak value and a real parameter.

The framework applies to any observable of the probe, not just eigenstates.

Abstract

A weak measurement consists in coupling a system to a probe in such a way that constructive interference generates a large output. So far, only the average output of the probe and its variance were studied. Here, the characteristic function for the moments of the output is provided. The outputs considered are not limited to the eigenstates of the pointer or of its conjugate variable, so that the results apply to any observable $\Hat{o}$ of the probe. Furthermore, a family of well behaved complex quantities, the normal weak values, is introduced, in terms of which the statistics of the weak measurement can be described. It is shown that, within a good approximation, the whole statistics of weak measurement is described by a complex parameter, the weak value, and a real one.