A Stronger Multi-observable Uncertainty Relation

TL;DR

This paper introduces a new variance-based sum uncertainty relation for multiple incompatible observables that provides a tighter lower bound than previous formulations, enhancing our understanding of quantum incompatibility.

Contribution

It proposes a stronger uncertainty relation for multiple observables that surpasses previous bounds, improving quantum measurement precision.

Findings

The new relation offers a better lower bound for spin-1/2 particles.

The relation is more effective for spin-1 particles.

It outperforms existing uncertainty bounds in various scenarios.

Abstract

Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables in the preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here we present a variance-based sum uncertainty relation for $N$ incompatible observables stronger than the simple generalization of the uncertainty relation for two observables derived by Maccone and Pati [Phys. Rev. Lett. {\bf113}, 260401 (2014)]. Further comparisons of our uncertainty relation with other related ones for spin-$\frac{1}{2}$ and spin-$1$ particles indicate that the obtained uncertainty relation gives a better lower bound.