Weak values under uncertain conditions

TL;DR

This paper investigates how averaging weak values over ensembles of pre- and post-selected states, including decoherence effects, constrains the weak value within the eigenvalue range of the measured observable.

Contribution

It provides a theoretical analysis of the behavior of weak values under statistical averaging and decoherence, clarifying their limits in uncertain conditions.

Findings

Averaging weak values reduces them within the eigenvalue spectrum.

Including decoherence via density matrices affects the averaged weak value.

Weak values can be constrained by statistical and environmental factors.

Abstract

We analyze the average of weak values over statistical ensembles of pre- and post-selected states. The protocol of weak values, proposed by Aharonov et al., is the result of a weak measurement conditional on the outcome of a subsequent strong (projective) measurement. Weak values can be beyond the range of eigenvalues of the measured observable and, in general, can be complex numbers. We show that averaging over ensembles of pre- and post-selected states reduces the weak value within the range of eigenvalues of the measured operator. We further show that the averaged result expressed in terms of pre- and post-selected density matrices, allows us to include the effect of decoherence.