  We prove a variety of new and refined uniform continuity bounds for entropies
of both classical random variables on an infinite state space and of quantum
states of infinite-dimensional systems. We obtain the first tight continuity
estimate on the Shannon entropy of random variables with a countably infinite
alphabet. The proof relies on a new mean-constrained Fano-type inequality and
the notion of maximal coupling of random variables. We then employ this
classical result to derive the first tight energy-constrained continuity bound
for the von Neumann entropy of states of infinite-dimensional quantum systems,
when the Hamiltonian is the number operator, which is arguably the most
relevant Hamiltonian in the study of infinite-dimensional quantum systems in
the context of quantum information theory.
  The above scheme works only for Shannon- and von Neumann entropies. Hence, to
deal with more general entropies, e.g. $\alpha$-R\'enyi and $\alpha$-Tsallis
entropies, with $\alpha \in (0,1)$, for which continuity bounds are known only
for finite-dimensional systems, we develop a novel approximation scheme which
relies on recent results on operator H\"older continuous functions and the
equivalence of all Schatten norms in special spectral subspaces of the
Hamiltonian. This approach is, as we show, motivated by continuity bounds for
$\alpha$-R\'enyi and $\alpha$-Tsallis entropies of random variables that follow
from the H\"older continuity of the entropy functionals. Bounds for $\alpha>1$
are provided, too. Finally, we settle an open problem on related approximation
questions posed in the recent works by Shirokov on the so-called
Finite-dimensional Approximation (FA) property.
