Triple measurements uncertainty and the distinguishment between the separable and entangled states

TL;DR

This work derives tight uncertainty relations for three observables in both sum and product forms using a variance-based method anchored to a global operator $R=\sum_{j=1}^3 H_j \otimes \sigma_j$. It proves a sum-form bound $\sum_j \Delta H_j^2 \ge \frac{1}{\sqrt{3}}\sum_j |\langle [H_j,H_{j+1}]\rangle|$ and, via a rescaling argument, a corresponding product-form bound $\prod_j \Delta H_j^2 \ge (\frac{1}{\sqrt{3}})^3 \prod_j |\langle [H_j,H_{j+1}]\rangle|$, with a physical interpretation tied to noncommutativity. The paper then links these uncertainty constants to entanglement detection, showing separable states have a positive variance lower bound while certain entangled eigenstates of $R$ can saturate or reduce the bound; it provides practical witnesses using $\langle R\rangle$ and illustrates the approach with a concrete example $H_j=\sigma_j$. Overall, the results extend entanglement detection criteria to observables lacking sufficient commutativity and give a physically transparent framework connecting uncertainty and entanglement for three-measurement scenarios.

Abstract

Uncertainty and entanglement are both profound and key concepts in quantum theory. For three observables, the tightest uncertainty constants for both product and summation forms are revealed. In this work, we give an alternative proof for three observables, also with a physical interpretation of the uncertainty constants. Our results show that such constants are intimately connected with the distinguishment between separable and entangled states.