Path probabilities for consecutive measurements, and certain "quantum paradoxes"

TL;DR

This paper analyzes quantum systems with intermediate measurements, showing how different measurement choices produce incompatible ensembles and paradoxes, and clarifies the role of weak measurements in revealing virtual paths without real trajectories.

Contribution

It provides a detailed framework for understanding how intermediate measurements define real and virtual paths, explaining quantum paradoxes through path probabilities in finite-dimensional systems.

Findings

Different measurement sets produce incompatible statistical ensembles.

Weak measurements do not reveal real paths, only limited information about virtual paths.

Quantum paradoxes can be understood through the analysis of path probabilities.

Abstract

We consider a finite-dimensional quantum system, making a transition between known initial and final states. The outcomes of several accurate measurements, which {\it could be} made in the interim, define virtual paths, each endowed with a probability amplitude. If the measurements are {\it actually made}, the paths, which may now be called "real", acquire also the probabilities, related to the frequencies, with which a path is seen to be travelled in a series of identical trials. Different sets of measurements, made on the same system, can produce different, or incompatible, statistical ensembles, whose conflicting attributes may, although by no means should, appear "paradoxical". We describe in detail the ensembles, resulting from intermediate measurements of mutually commuting, or non-commuting, operators, in terms of the real paths produced. In the same manner, we analyse the Hardy's and the "three box" paradoxes, the photon's past in an interferometer, the "quantum Cheshire cat" experiment, an the closely related subject of "interaction-free measurements". It is shown that, in all these cases, inaccurate "weak measurements" produce no real paths, and yield only limited information about the virtual paths' probability amplitudes