Weak measurement takes a simple form for cumulants

TL;DR

This paper demonstrates that cumulants of pointer positions in weak measurements are directly proportional to cumulants of sequential weak values, simplifying the understanding of complex relationships involving initial pointer wavefunctions.

Contribution

It reveals that cumulants provide a fundamental and simplified link between pointer measurements and weak values, even with complex initial pointer states.

Findings

Cumulants of pointer positions are proportional to cumulants of sequential weak values.

The complexity of pointer wavefunctions simplifies when using cumulants.

Cumulants have fundamental physical significance in weak measurement theory.

Abstract

A weak measurement on a system is made by coupling a pointer weakly to the system and then measuring the position of the pointer. If the initial wavefunction for the pointer is real, the mean displacement of the pointer is proportional to the so-called weak value of the observable being measured. This gives an intuitively direct way of understanding weak measurement. However, if the initial pointer wavefunction takes complex values, the relationship between pointer displacement and weak value is not quite so simple, as pointed out recently by R. Jozsa. This is even more striking in the case of sequential weak measurements. These are carried out by coupling several pointers at different stages of evolution of the system, and the relationship between the products of the measured pointer positions and the sequential weak values can become extremely complicated for an arbitrary initial pointer wavefunction. Surprisingly, all this complication vanishes when one calculates the cumulants of pointer positions. These are directly proportional to the cumulants of sequential weak values. This suggests that cumulants have a fundamental physical significance for weak measurement.