Rényi formulation of uncertainty relations for POVMs assigned to a quantum design

Abstract: Information entropies provide powerful and flexible way to express restrictions imposed by the uncertainty principle. This approach seems to be very suitable in application to problems of quantum information theory. It is typical that questions of such a kind involve measurements having one or another specific structure. The latter often allows us to improve entropic bounds that follow from uncertainty relations of sufficiently general scope. Quantum designs have found use in many issues of quantum information theory, whence uncertainty relations for related measurements are of interest. In this paper, we obtain uncertainty relations in terms of min-entropies and R\'{e}nyi entropies for POVMs assigned to a quantum design. Relations of the Landau--Pollak type are addressed as well. Using examples of quantum designs in two dimensions, the obtained lower bounds are then compared with the previous ones. An impact on entropic steering inequalities is briefly discussed.