Stronger uncertainty relations with improvable upper and lower bounds

Abstract

We utilize quantum superposition principle to establish the improvable upper and lower bounds on the stronger uncertainty relation, i.e., the "weighted-like" sum of the variances of observables. Our bounds include some free parameters which not only guarantee the nontrivial bounds but also can effectively control the bounds as tightly as one expects. Especially, these parameters don't obviously depend on the state and observables. It also implies one advantage of our method that any nontrivial bound can always be more improvable. In addition, we generalize both bounds to the uncertainty relation with multiple observables, but the perfect tightness is not changed. Examples are given to illustrate the improvability of our bounds in each case.