Notes on Entropic Uncertainty Relations Beyond the Scope of Riesz’s Theorem

Abstract

We discuss those relations for entropies of two quantum measurements that do not follow from the Riesz theorem and its varieties. As measures of an uncertainty, both the Renyi entropies and Tsallis entropies are utilized. It is assumed that values of entropic parameters do not share the usual relation which is required for application of Riesz’s theorem. As is shown, the question is quite reduced to estimating from below the sum or the product of norm-like functions of two generated probability distributions. The considered approach is developed in details within the two examples. In the first example, entropic uncertainty relations are given for a pair of spin-1/2 components along two non-orthogonal axes. The second example deals with two measurements for state discrimination. The former projective measurement is assigned to the Helstrom scheme, the second POVM is related to the B92 protocol of quantum key distribution.