  We present a bouquet of continuity bounds for quantum entropies, falling
broadly into two classes: First, a tight analysis of the Alicki-Fannes
continuity bounds for the conditional von Neumann entropy, reaching almost the
best possible form that depends only on the system dimension and the trace
distance of the states. Almost the same proof can be used to derive similar
continuity bounds for the relative entropy distance from a convex set of states
or positive operators. As applications we give new proofs, with tighter bounds,
of the asymptotic continuity of the relative entropy of entanglement, $E_R$,
and its regularization $E_R^\infty$, as well as of the entanglement of
formation, $E_F$. Using a novel "quantum coupling" of density operators, which
may be of independent interest, we extend the latter to an asymptotic
continuity bound for the regularized entanglement of formation, aka
entanglement cost, $E_C=E_F^\infty$.
  Second, analogous continuity bounds for the von Neumann entropy and
conditional entropy in infinite dimensional systems under an energy constraint,
most importantly systems of multiple quantum harmonic oscillators. While
without an energy bound the entropy is discontinuous, it is well-known to be
continuous on states of bounded energy. However, a quantitative statement to
that effect seems not to have been known. Here, under some regularity
assumptions on the Hamiltonian, we find that, quite intuitively, the Gibbs
entropy at the given energy roughly takes the role of the Hilbert space
dimension in the finite-dimensional Fannes inequality.
