  This dissertation investigates relative entropies, also called generalized
divergences, and how they can be used to characterize information-theoretic
tasks in quantum information theory. The main goal is to further refine
characterizations of the optimal rates for quantum source coding, state
redistribution, and measurement compression with quantum side information via
second order asymptotic expansions and strong converse theorems. The
dissertation consists of a mathematical and an information-theoretic part. In
the mathematical part, we focus on the $\alpha$-sandwiched R\'enyi divergence
($\alpha$-SRD). We first investigate the limit $\alpha\to 0$ to determine
whether this recovers the well-known $0$-R\'enyi relative divergence. We then
prove various new results for entropic quantities derived from the
$\alpha$-SRD, including dimension bounds and useful bounds in terms of the
fidelity between two quantum states. Furthermore, we derive a necessary and
sufficient algebraic condition for equality in the data processing inequality
(viz. monotonicity under quantum operations) for the $\alpha$-SRD, and give
applications to entropic bounds. In the information-theoretic part, we first
derive the second order asymptotics of visible quantum source coding using a
mixed source. For the achievability part, we develop universal quantum source
codes achieving a given second order rate for a memoryless source. As a
corollary of the main result, we obtain the second order asymptotics of quantum
source coding using a single memoryless source. We then prove strong converse
theorems for state redistribution (with or without feedback) and measurement
compression with quantum side information. The key ingredients in proving these
theorems are the aforementioned fidelity bounds on R\'enyi entropic quantities
derived from the $\alpha$-SRD.
