  Recently a new quantum generalization of the Renyi divergence and the
corresponding conditional Renyi entropies was proposed. Here we report on a
surprising relation between conditional Renyi entropies based on this new
generalization and conditional Renyi entropies based on the quantum relative
Renyi entropy that was used in previous literature. Our result generalizes the
well-known duality relation H(A|B) + H(A|C) = 0 of the conditional von Neumann
entropy for tripartite pure states to Renyi entropies of two different kinds.
  As a direct application, we prove a collection of inequalities that relate
different conditional Renyi entropies and derive a new entropic uncertainty
relation.
