# Optimal joint measurements of complementary observables by a single trapped ion

## TL;DR

This paper experimentally demonstrates the optimal joint measurement of complementary observables using a single trapped ion, confirming a new error trade-off relation and advancing precision measurement in quantum information science.

## Contribution

It provides the first direct test of a new error trade-off relation for compatible observables approximating incompatible ones using a single trapped ion.

## Key findings

- Confirmed tight bounds of the error trade-off relation.
- Demonstrated optimal joint measurements with a single ultracold ion.
- Potential applications in quantum information and high-precision measurement.

## Abstract

The uncertainty relations, pioneered by Werner Heisenberg nearly 90 years ago, set a fundamental limitation on the joint measurability of complementary observables. This limitation has long been a subject of debate, which has been reignited recently due to new proposed forms of measurement uncertainty relations. The present work is associated with a new error trade-off relation for compatible observables approximating two incompatible observables, in keeping with the spirit of Heisenberg's original ideas of 1927. We report the first \textsl{direct} test and confirmation of the tight bounds prescribed by such an error trade-off relation, based on an experimental realisation of optimal joint measurements of complementary observables using a single ultracold $^{40}Ca^{+}$ ion trapped in a harmonic potential. Our work provides a prototypical determination of ultimate joint measurement error bounds with potential applications in quantum information science for high-precision measurement and information security.