Weighted Uncertainty Relations

TL;DR

This paper introduces a family of weighted uncertainty relations that provide optimal lower bounds for variances in quantum systems, applicable to multiple observables and removing previous restrictions on quantum states.

Contribution

It develops a generalized framework for weighted uncertainty relations that improve upon existing bounds and extend to multi-observable scenarios.

Findings

Provides a family of weighted uncertainty relations with optimal bounds

Removes restrictions on quantum states for uncertainty relations

Generalizes to multiple observables with optimal bounds

Abstract

Recently, Maccone and Pati have given two stronger uncertainty relations based on the sum of variances and one of them is nontrivial when the quantum state is not an eigenstate of the sum of the observables. We derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations and remove the restriction on the quantum state. Generalization to multi-observable cases is also given and an optimal lower bound for the weighted sum of the variances is obtained in general quantum situation.