Abstract
In the course of the last decades entropic uncertainty relations have attracted much attention not only due to their fundamental role as manifestation of non-classicality 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 work 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.
Highlights
Quantum mechanics prohibits observers to make simultaneous predictions about complementary properties of a physical system [1]
We have proven bounds of entropic uncertainty relations in terms of Rényi and Tsallis entropies for measurements whose effects form quantum t-designs
We exploited the fact that quantum t-designs form projectors onto the symmetric subspace of a t-fold tensor product space which allowed us to make use of elements of representation theory of the symmetric group
Summary
Quantum mechanics prohibits observers to make simultaneous predictions about complementary properties of a physical system [1] This fundamental aspect manifests itself through restrictions of uncertainties of measurement outcomes performed on a number of identically prepared copies of a system and is strongly related to the noncommuting structure of the observables under consideration. By invoking the properties of observables with mutually unbiased eigenbases it was possible to derive EURs involving more than two observables [26] The latter have been first obtained for Shannon entropies [27,28], and were later generalized to Rényi and Tsallis entropies [29,30,31,32,33]. We exploit the pseudorandom properties of the effects representing the measurements under considerations Whenever the latter form a so-called quantum design, we are able to prove lower bounds on the sums of the respective generalized entropies.
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