Optimal Universal Uncertainty Relations
  We study universal uncertainty relations and present a method called joint probability distribution diagram to improve the majorization bounds constructed independently in [Phys. Rev. Lett. 111, 230401 (2013)] and [J. Phys. A. 46, 272002 (2013)]. The results give rise to state independent uncertainty relations satisfied by any nonnegative Schur-concave functions. On the other hand, a remarkable recent result of entropic uncertainty relation is the direct-sum majorization relation. In this paper, we illustrate our bounds by showing how they provide a complement to that in [Phys. Rev. A. 89, 052115 (2014)]. 
Tao Li, Yunlong Xiao, Teng Ma, Shao-Ming Fei, Naihuan Jing, Xianqing   Li-Jost, and Zhi-Xi Wang
quant-ph