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, 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.

Experimental investigation of the uncertainty relations with coherent light

“…Our results on the variance-based uncertainty relations of equations (16)(17) agrees with the predictions of quantum mechanics, and also with previous measurements using single photon sources, e.g. see Figures 2 and 3 in Ref.…”

Entropic uncertainty relation based on generalized uncertainty principle

“…It is obviously important to explore the analytic form of the correction to understand the property of quantum gravity through entropic entropy. For example, in a finite dimensional system, an explicit map between a variance type uncertainty relation and an entropic type of it have been constructed [21]. It is then intriguing to investigate a similar type of mappings in our case.…”

Multipartite steering inequalities based on entropic uncertainty relations

“…However, here we consider only the case where there is no entanglement between Bob and Charlie, since we are interested in detecting the possible simplest form of these quantum correlations. If the state is non-GMS then the ensemble (27) can be expressed as:…”