# Number and phase: complementarity and joint measurement uncertainties

## TL;DR

This paper demonstrates the complementarity of number and phase in a single mode optical field and establishes bounds on the measurement uncertainties for their approximate joint measurements.

## Contribution

It introduces a formal analysis of the complementarity between number and phase and derives bounds on their joint measurement uncertainties.

## Key findings

- Number and phase are complementary observables.
- Bounds on measurement uncertainty regions are established.
- Provides a framework for approximate joint measurements.

## Abstract

We show that number and canonical phase (of a single mode optical field) are complementary observables. We also bound the measurement uncertainty region for their approximate joint measurements.