Stronger Reverse Uncertainty Relation for Multiple Incompatible Observables

TL;DR

This paper introduces a new reverse uncertainty relation that overcomes previous limitations, unifies it with the normal uncertainty relation, and extends it to multiple observables with adjustable tightness, with applications in purity detection.

Contribution

The authors develop a refined reverse uncertainty relation that fixes the infinity issue, unifies it with the standard relation, and generalizes it to multiple observables with customizable tightness.

Findings

The new relation avoids the infinity problem present in previous bounds.

It demonstrates the equivalence between reverse and normal uncertainty relations.

Application in purity detection shows practical utility.

Abstract

Recently,D.Mondal et.al[Phys. Rev. A. 95, 052117(2017)]creatively introduce a new interesting concept of reverse uncertainty relation which indicates that one cannot only prepare quantum states with joint small uncertainty, but also with joint great uncertainty for incompatible observables. However, the uncertainty upper bound they constructed cannot express the essence of this concept well, i.e., the upper bound will go to infinity in some cases even for incompatible observables. Here, we construct a new reverse uncertainty relation and successfully fix this "infinity" problem. Also, it is found that the reverse uncertainty relation and the normal uncertainty relation are the same in essential, and they both can be unified by the same theoretical framework. Moreover, taking advantage of this unified framework, one can construct a reverse uncertainty relation for multiple observables with any tightness required. Meanwhile, the application of the new uncertainty relation in purity detection is discussed.