Strange Weak Values

TL;DR

This paper develops a formal framework for understanding weak values in quantum mechanics, focusing on their probabilistic interpretation, conditions for negative and strange values, and applications to Hardy's paradox and spin systems.

Contribution

It introduces a consistent formal theory of weak values, clarifies conditions for strange weak values, and applies the framework to well-known quantum paradoxes.

Findings

Negative weak values occur under specific post-selection conditions.

Strange weak values can be explained through counter-factual reasoning.

The framework clarifies the emergence of strange weak values in quantum paradoxes.

Abstract

We develop a formal theory of the weak values with emphasis on the consistency conditions and a probabilistic interpretation in the counter-factual processes. We present the condition for the choice of the post-selected state to give a negative weak value of a given projection operator and strange values of an observable in general. The general framework is applied to Hardy's paradox and the spin $1/2$ system to explicitly address the issues of counter-factuality and strange weak values. The counter-factual arguments which characterize the paradox specifies the pre-selected state and a complete set of the post-selected states clarifies how the strange weak values emerge.