Entropic uncertainty relations from quantum designs

Abstract

In the course of the past decades entropic uncertainty relations have attracted much attention not only due to their fundamental role as manifestation of nonclassicality of quantum mechanics, but also as major tools for applications of quantum information theory. Amongst the latter are protocols for the detection of quantum correlations or for the secure distribution of secret keys. In this article we show how to derive entropic uncertainty relations for sets of measurements whose effects form quantum designs. The key property of quantum designs is their indistinguishability from truly random quantum processes as long as one is concerned with moments up to some finite order. Exploiting this characteristic enables us to evaluate polynomial functions of measurement probabilities which leads to lower bounds on sums of generalized entropies. As an application we use the derived uncertainty relations to investigate the incompatibility of sets of binary observables.