On the Validity of the Weak Value Approximation

TL;DR

This paper addresses the lack of a rigorous derivation for the weak value approximation by presenting a convergence-based derivation of the weak-value form under weak coupling, validated through two concrete probe models: the von Neumann Gaussian-pointer and a two-state (qubit) probe. It introduces a normalized postselected probe state to manage nonunitary effects from potentially complex weak values and constructs bounds that guarantee the exact post-interaction state converges to the weak-value form as $\epsilon \to 0$, with explicit results for both probe models. The work provides detailed calculations of probe-state evolution, including the shift in the von Neumann pointer and the joint real/imaginary-part dependencies in the qubit case, and establishes systematic criteria (via $\bar a$, $\bar w$, $\epsilon$, and probe spread $\Delta$) for controlling the error terms. By delivering a rigorous mathematical foundation and a generalizable framework, the paper strengthens the validity and applicability of weak value techniques and sets the stage for extending the method to additional models and observables. The contributions thus offer a robust, transferable approach to validating weak-value approximations with potential implications for weak-value amplification and foundational interpretations in quantum measurement.

Abstract

The weak value approximation has been in use for thirty-five years, but it has not as of yet received a truly complete derivation, leaving its mathematical validity in a state of limbo. Herein, I fill this gap, deriving the weak value approximation under the von Neumann and qubit probe models. Not only does this provide a level of mathematical support to the weak value approximation not attained in previous works, but the techniques demonstrated in the process might be usable by others to forge similar derivations for alternative models, thus teasing the possibility of even broader validation in the future.