Quantum measurements, stochastic networks, the uncertainty principle, and not so strange "weak values"

TL;DR

This paper explores how quantum measurement outcomes can be modeled as stochastic networks, linking the uncertainty principle to the behavior of weak values and the nature of quantum paths.

Contribution

It introduces a framework connecting quantum measurement outcomes with stochastic networks and clarifies the role of the uncertainty principle in weak values.

Findings

Quantum outcomes form stochastic networks of possible paths.

Accurate measurements make virtual paths 'real' and relate mean values to frequencies.

Weak values depend on pre- and post-selected states and are influenced by interference.

Abstract

The outcomes of a series of measurements, made on a quantum system, form a sequence of random events which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become "real", and the mean values of a quantity (a functional) is directly related to the frequencies with which the paths are travelled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one to distinguish between interfering alternatives, while leaving the interference between them intact.