Multi-observable Uncertainty Equality based on the sum of standard deviations in the qubit system

TL;DR

This paper introduces a new multi-observable uncertainty equality and inequality for qubit systems, improving the accuracy of uncertainty relations and their ability to detect mixedness and incompatibility.

Contribution

It presents a novel uncertainty equality and a tighter inequality based on the sum of standard deviations, enhancing the precision of uncertainty relations in qubit systems.

Findings

Uncertainty equality accurately expresses the relation in qubit systems.

New inequality provides a tighter lower bound even in open systems.

Fixes the issue of null lower bounds in product form uncertainty relations.

Abstract

We construct a multi-observable uncertainty equality as well as an inequality based on the sum of standard deviations in the qubit system. The obtained equality indicates that the uncertainty relation can be expressed more accurately, and also can be used to detect the mixedness of the system. Meanwhile, the new uncertainty inequality can provide a tighter lower bound, and the tightness can be maintained at a high level even in an open system. Furthermore, the deficiency of the uncertainty relation, that the lower bound of the product form uncertainty relations can be null even for two incompatible observables, can be completely fixed by the new uncertainty relation.