  Quantum generalizations of R\'enyi's entropies are a useful tool to describe
a variety of operational tasks in quantum information processing. Two families
of such generalizations turn out to be particularly useful: the Petz quantum
R\'enyi divergence $\bar{D}_{\alpha}$ and the minimal quantum R\'enyi
divergence $\widetilde{D}_{\alpha}$. Moreover, the maximum quantum R\'enyi
divergence $\widehat{D}_{\alpha}$ is of particular mathematical interest. In
this Master thesis, we investigate relations between these divergences and
their applications in quantum information theory. Our main result is a reverse
Araki-Lieb-Thirring inequality that implies a new relation between the minimal
and the Petz divergence, namely that $\alpha \bar{D}_{\alpha}(\rho \| \sigma)
\leqslant \widetilde{D}_{\alpha}(\rho \| \sigma)$ for $\alpha \in [0,1]$ and
where $\rho$ and $\sigma$ are density operators. This bound suggests defining a
"pretty good fidelity", whose relation to the usual fidelity implies the known
relations between the optimal and pretty good measurement as well as the
optimal and pretty good singlet fraction. In addition, we provide a new proof
of the inequality $\widetilde{D}_{1}(\rho \| \sigma) \leqslant
\widehat{D}_{1}(\rho \| \sigma)\, ,$ based on the Araki-Lieb-Thirring
inequality. This leads to an elegant proof of the logarithmic form of the
reverse Golden-Thompson inequality.
