  In the first part of this thesis, we present a general technique for
establishing local and uniform continuity bounds for Schur concave functions.
Our technique uses a particular relationship between majorization and the trace
distance between quantum states. Namely, the majorization pre-order attains a
minimum over $\epsilon$-balls in this distance. By tracing the path of the
majorization-minimizer as a function of the distance $\epsilon$, we obtain the
path of "majorization flow". This yields a new proof of the Audenaert-Fannes
continuity bound for the von Neumann entropy in a universal framework which
extends to the other functions, including the $\alpha$-R\'enyi entropy, for
which we obtain novel bounds in the case $\alpha > 1$. We apply this technique
to other Schur concave functions, such as the number of connected components of
a certain random graph model, and the number of distinct realizations of a
random variable.
  In the second part, we consider repeated interaction systems, in which a
system of interest interacts with a sequence of probes one at a time. We
characterize which repeated interaction systems break any initially-present
entanglement between the system and an untouched reference after finitely many
steps. Additionally, when the probes and their interactions with the system are
slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound,
an inequality between the entropy change of the system and the energy change of
the probes, in the limit in which the number of steps tends to infinity and
both the difference between consecutive probes and the difference between their
interactions vanishes. This analysis proceeds at a fine-grained level by means
of a two-time measurement protocol, in which the energy of the probes is
measured before and after each interaction.
