# Complementary observables in quantum mechanics

## TL;DR

This paper reviews the concept of complementarity of observables in quantum mechanics, clarifies its operational meaning, introduces new characterisations, and explores applications including position-momentum and interferometry, with considerations of noise effects.

## Contribution

It provides new unified characterisations of quantum complementarity based on a factorisation lemma, enhancing understanding of operational and noisy aspects.

## Key findings

- New characterisations of complementarity in quantum effects
- Application to position-momentum and interferometry
- Analysis of complementarity in noisy, convoluted settings

## Abstract

We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity - some of which new - in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position-momentum, position-energy, number-phase, as well as periodic observables relevant to spatial interferometry. We close the paper with some considerations of complementarity in a noisy setting, focusing especially on the case of convolutions of position and momentum, which was a recurring topic in Paul's work on operational formulation of quantum measurements and central to his philosophy of unsharp reality.