Understanding weak values without new probability theory

TL;DR

This paper explains the physical meaning of weak values and measurements using standard probability theory and the Born rule, clarifying why weak values can lie outside the eigenvalue spectrum.

Contribution

It demonstrates that understanding weak values does not require new probability theory, but can be achieved through existing frameworks and the Born rule.

Findings

Weak values can be interpreted as expectation values conditioned on post-selection.

Weak values can lie outside the eigenvalue range of the observable.

Standard probability theory suffices to understand weak measurement phenomena.

Abstract

The physical meaning of weak values and measurements can be completely understood with Born rule and the general probability theory. It is known that the weak value of an observable $\hat A$ with post-selection $\langle F|$ may be out of the eigenvalue range of $\hat A$. This is because the weak value of $\hat A$ with the post-selection is, in general, not the expectation value of $\hat A$, but the expectation value of $\hat A| F\rangle\langle F|$ boosted by the post-selection.