  We consider two fundamental tasks in quantum information theory, data
compression with quantum side information as well as randomness extraction
against quantum side information. We characterize these tasks for general
sources using so-called one-shot entropies. We show that these
characterizations - in contrast to earlier results - enable us to derive tight
second order asymptotics for these tasks in the i.i.d. limit. More generally,
our derivation establishes a hierarchy of information quantities that can be
used to investigate information theoretic tasks in the quantum domain: The
one-shot entropies most accurately describe an operational quantity, yet they
tend to be difficult to calculate for large systems. We show that they
asymptotically agree up to logarithmic terms with entropies related to the
quantum and classical information spectrum, which are easier to calculate in
the i.i.d. limit. Our techniques also naturally yields bounds on operational
quantities for finite block lengths.
