Direct experimental test of forward and reverse uncertainty relations

Lei Xiao,Arun Kumar Pati,Bowen Fan,Peng Xue,Kunkun Wang

doi:10.1103/physrevresearch.2.023106

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.

Highlights

Uncertainty relations [1,2,3,4,5] are the hallmarks of quantum physics and have been widely investigated since their inception [6,7,8,9,10,11,12,13,14,15,16,17]
We experimentally test multifold of statedependent 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-Schrödinger uncertainty relation and other ones found in current literatures
We report an experimental test of these new uncertainty relations with optimization free bounds and reverse uncertainty relations for single-photon measurements and demonstrate they are valid for states of a spin-1 particle

Summary

INTRODUCTION

Uncertainty relations [1,2,3,4,5] are the hallmarks of quantum physics and have been widely investigated since their inception [6,7,8,9,10,11,12,13,14,15,16,17] These uncertainty relations impose a fundamental limitation on the possible preparation of quantum states for which two noncommuting observables can have sharp values—often refereed to as “preparation” uncertainty relations. State-dependent uncertainty relations with the optimization free uncertainty bounds both in the sum and the product forms were derived by Mondal et al in [5]. Our test realizes a direct measurement model which leverages the requirement of quantum state tomography [9,15,29]

THEORETICAL FRAMEWORK

EXPERIMENTAL INVESTIGATION

EXPERIMENTAL RESULTS

CONCLUSION