Direct Experimental Test of Forward and Reverse Uncertainty Relations

TL;DR

This paper experimentally tests new, tighter forward and reverse uncertainty relations for incompatible observables in photonic qutrits, providing deeper insight into quantum preparation uncertainty and measurement precision limits.

Contribution

It provides the first experimental verification of state-dependent reverse uncertainty relations, demonstrating their tightness and independence from optimization.

Findings

Tight forward and reverse uncertainty relations confirmed experimentally.

Reverse relations reveal new aspects of preparation uncertainty.

Results suggest potential applications in quantum measurement precision.

Abstract

The canonical Robertson-Schr\"{o}dinger uncertainty relation provides a loose bound for the product of variances of two non-commuting observables. Recently, several tight forward and reverse uncertainty relations have been proved which go beyond the traditional uncertainty relations. Here we experimentally test multifold of state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables for photonic qutrits. These uncertainty relations are independent of any optimization and still tighter than the Robertson-Schr\"{o}dinger uncertainty relation and other ones found in current literatures. For the first time, we also test the state-dependent reverse uncertainty relations for the sum of variances of two incompatible observables, which implies another unique feature of preparation uncertainty in quantum mechanics. Our experimental results not only foster insight into a fundamental limitation on preparation uncertainty but also may contribute to the study of upper limit of precession measurements for incompatible observables.