Weak values, 'negative probability' and the uncertainty principle

TL;DR

This paper explores the nature of weak values in quantum mechanics, showing they can be negative and do not necessarily reflect the actual range of quantum variables, affecting interpretation of weak measurements.

Contribution

It provides a detailed analysis of weak values, revealing their potential negativity and unusual properties, and discusses implications for quantum measurement interpretation.

Findings

Weak values can be negative, not reflecting the true variable range.

Moments of amplitude distributions have unusual properties.

Weak measurement results can be misinterpreted due to these properties.

Abstract

A quantum transition can be seen as a result of interference between various pathways(e.g. Feynman paths) which can be labelled by a variable $f$. An attempt to determine the value of f without destroying the coherence between the pathways produces a weak value of $\bar{f}$. We show $\bar{f}$ to be an average obtained with amplitude distribution which can, in general, take negative values which, in accordance with the uncertainty principle, need not contain information about the actual range of the values $f$ which contribute to the transition. It is also demonstrated that the moments of such alternating distributions have a number of unusual properties which may lead to misinterpretation of the weak measurement results.We provide a detailed analysis of weak measurements with and without post-selection. Examples include the double slit diffraction experiment,weak von Neumann and von Neumann-like measurements, traversal time for an elastic collision, the phase time, the local angular momentum(LAM) and the 'three-box case' of {\it Aharonov et al}