Weak values are universal in von Neumann measurements

TL;DR

This paper demonstrates that weak values are a universal feature in von Neumann measurements, applicable beyond weak measurement regimes, by expressing detector averages through generalized weak values for any coupling strength.

Contribution

It shows that weak values are not exclusive to weak measurements and provides a unified framework describing detector averages for all coupling strengths and initial states.

Findings

Detector averages are described by generalized weak values for any coupling strength.

Higher-order detector moments also have weak value expansions.

Expressions simplify for Hermite-Gauss mode detectors, linking to single system weak values.

Abstract

We refute the widely held belief that the quantum weak value necessarily pertains to weak measurements. To accomplish this, we use the transverse position of a beam as the detector for the conditioned von Neumann measurement of a system observable. For any coupling strength, any initial states, and any choice of conditioning, the averages of the detector position and momentum are completely described by the real parts of three generalized weak values in the joint Hilbert space. Higher-order detector moments also have similar weak value expansions. Using the Wigner distribution of the initial detector state, we find compact expressions for these weak values within the reduced system Hilbert space. As an application of the approach, we show that for any Hermite-Gauss mode of a paraxial beam-like detector these expressions reduce to the real and imaginary parts of a single system weak value plus an additional weak-value-like contribution that only affects the momentum shift.