In this Thesis, several results in quantum information theory are collected,
most of which use entropy as the main mathematical tool. *While a direct
generalization of the Shannon entropy to density matrices, the von Neumann
entropy behaves differently. A long-standing open question is, whether there
are quantum analogues of unconstrained non-Shannon type inequalities. Here, a
new constrained non-von-Neumann type inequality is proven, a step towards a
conjectured unconstrained inequality by Linden and Winter. *IID quantum state
merging can be optimally achieved using the decoupling technique. The one-shot
results by Berta et al. and Anshu at al., however, had to bring in additional
mathematical machinery. We introduce a natural generalized decoupling paradigm,
catalytic decoupling, that can reproduce the aforementioned results when used
analogously to the application of standard decoupling in the asymptotic case.
*Port based teleportation, a variant of standard quantum teleportation
protocol, cannot be implemented perfectly. We prove several lower bounds on the
necessary number of output ports N to achieve port based teleportation for
given error and input dimension, showing that N diverges uniformly in the
dimension of the teleported quantum system, for vanishing error. As a
byproduct, a new lower bound for the size of the program register for an
approximate universal programmable quantum processor is derived. *In the last
part, we give a new definition for information-theoretic quantum
non-malleability, strengthening the previous definition by Ambainis et al. We
show that quantum non-malleability implies secrecy, analogous to quantum
authentication. Furthermore, non-malleable encryption schemes can be used as a
primitive to build authenticating encryption schemes. We also show that the
strong notion of authentication recently proposed by Garg et al. can be
fulfilled using 2-designs.