Sum uncertainty relations for arbitrary N incompatible observables.

Bin Chen,Shao-Ming Fei

doi:10.1038/srep14238

Abstract

We formulate uncertainty relations for arbitrary N observables. Two uncertainty inequalities are presented in terms of the sum of variances and standard deviations, respectively. The lower bounds of the corresponding sum uncertainty relations are explicitly derived. These bounds are shown to be tighter than the ones such as derived from the uncertainty inequality for two observables [Phys. Rev. Lett. 113, 260401 (2014)]. Detailed examples are presented to compare among our results with some existing ones.

Highlights

We formulate uncertainty relations for arbitrary N observables
The corresponding uncertainty inequalities are of great importance for both theoretical investigation and experimental implementation
We present a sum of variance-based uncertainty relation and a standard deviation-based sum uncertainty relation for N observables

Summary

Introduction

We formulate uncertainty relations for arbitrary N observables. Two uncertainty inequalities are presented in terms of the sum of variances and standard deviations, respectively. The product of the standard deviation ΔAΔB is null if the measured state ψ is an eigenstate of one of the two observables. It is shown that the lower bounds of their uncertainty inequalities are nontrivial, whenever the two observables are incompatible with respect to the measured states (the states are not common eigenstates of both two observables).

Results

Conclusion