Error and unsharpness in approximate joint measurements of position and momentum

TL;DR

This paper reviews recent advances in quantifying measurement error in quantum mechanics, focusing on the trade-offs and criteria for approximate joint measurements of position and momentum.

Contribution

It provides a comparative analysis of various error measures and evaluates their effectiveness as criteria for good quantum measurement approximations.

Findings

Different error measures have distinct properties and suitability.

Heisenberg's measurement uncertainty relation can be precisely formulated.

Trade-offs exist between measurement errors of incompatible observables.

Abstract

In recent years, novel quantifications of measurement error in quantum mechanics have for the first time enabled precise formulations of Heisenberg's famous but often challenged measurement uncertainty relation. These relations take the form of a trade-off for the necessary errors in joint approximate measurements of position and momentum and other incompatible pairs of observables. Here we review some of these error measures, examine their properties and suitability, and compare their relative strengths as criteria for "good" approximations.