Equivalence theorem of uncertainty relations

Abstract

We present an equivalence theorem to unify the two classes of uncertainty relations, i.e. the variance-based ones and the entropic forms, showing that the entropy of an operator in a quantum system can be built from the variances of a set of commutative operators. This means that 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 are obtained for the qubit system that are stronger than the existing ones in the literature, and variance-based uncertainty relations for spin systems are reached from the corresponding entropic uncertainty relations.