title: Weighted Uncertainty Relations.
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.
authors.affiliation: School of Mathematics, South China University of Technology, Guangzhou 510640, China.|Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany.
authors.forename: Yunlong
authors.initials: Y
authors.lastname: Xiao
authors.affiliation: School of Mathematics, South China University of Technology, Guangzhou 510640, China.|Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA.
authors.forename: Naihuan
authors.initials: N
authors.lastname: Jing
authors.affiliation: Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany.
authors.forename: Xianqing
authors.initials: X
authors.lastname: Li-Jost
authors.affiliation: Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany.|School of Mathematical Sciences, Capital Normal University, Beijing 100048, China.
authors.forename: Shao-Ming
authors.initials: SM
authors.lastname: Fei