Stronger uncertainty relations with improvable upper and lower bounds

TL;DR

This paper introduces a method to derive adjustable upper and lower bounds on quantum uncertainty relations, enhancing their tightness and applicability through free parameters that are largely independent of specific states or observables.

Contribution

The authors develop improvable bounds on quantum uncertainty relations using free parameters, generalize to multiple observables, and demonstrate their effectiveness with examples.

Findings

Bounds can be made arbitrarily tight

Parameters are largely independent of states and observables

Method applies to multiple observables

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.