  The Renyi entropies constitute a family of information measures that
generalizes the well-known Shannon entropy, inheriting many of its properties.
They appear in the form of unconditional and conditional entropies, relative
entropies or mutual information, and have found many applications in
information theory and beyond. Various generalizations of Renyi entropies to
the quantum setting have been proposed, most notably Petz's quasi-entropies and
Renner's conditional min-, max- and collision entropy. Here, we argue that
previous quantum extensions are incompatible and thus unsatisfactory.
  We propose a new quantum generalization of the family of Renyi entropies that
contains the von Neumann entropy, min-entropy, collision entropy and the
max-entropy as special cases, thus encompassing most quantum entropies in use
today. We show several natural properties for this definition, including
data-processing inequalities, a duality relation, and an entropic uncertainty
relation.
