  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)].
