Equivalence theorem of uncertainty relations

TL;DR

This paper introduces an equivalence theorem that unifies variance-based and entropic uncertainty relations in quantum mechanics, enabling translation between the two forms and deriving stronger or new uncertainty relations.

Contribution

The paper presents a novel equivalence theorem linking entropy and variance-based uncertainty relations, allowing for unified formulations and improved bounds in quantum uncertainty analysis.

Findings

Derived stronger entropic uncertainty relations for qubit systems.

Obtained variance-based uncertainty relations from entropic forms for spin systems.

Established a unified framework connecting different uncertainty relation types.

Abstract

We present an equivalence theorem to unify the two classes of uncertainty relations, i.e., the variance-based ones and the entropic forms, which shows that the entropy of an operator in a quantum system can be built from the variances of a set of commutative operators. That means an uncertainty relation in the language of entropy may be mapped onto a variance-based one, and vice versa. Employing the equivalence theorem, alternative formulations of entropic uncertainty relations stronger than existing ones in the literature are obtained for qubit system, and variance based uncertainty relations for spin systems are reached from the corresponding entropic uncertainty relations.