Variance-based uncertainty relation for incompatible observers

TL;DR

This paper introduces a new variance-based uncertainty relation for single-qubit systems, linking mixedness and measurement incompatibility, with potential applications in quantum control and information processing.

Contribution

It presents a novel uncertainty equality and inequality based on mixedness, providing a new way to quantify uncertainty and system mixedness in quantum systems.

Findings

Derived a new uncertainty equality for single-qubit systems.

Established a variance-based inequality related to mixedness.

Demonstrated the relation using feedback-controlled qubit systems.

Abstract

Based on mixedness definition as M=1-tr(\r{ho}^2), we obtain a new variance-based uncertainty equality along with an inequality for Hermitian operators of a single-qubit system. The obtained uncertainty equality can be used as a measure of the system mixedness. A qubit system with feedback control is also exploited to demonstrate the new uncertainty.