  Various quantum analogues of the central limit theorem, which is one of the
cornerstones of probability theory, are known in the literature. One such
analogue, due to Cushen and Hudson, is of particular relevance for quantum
optics. It implies that the state in any single output arm of an $n$-splitter,
which is fed with $n$ copies of a centred state $\rho$ with finite second
moments, converges to the Gaussian state with the same first and second moments
as $\rho$. Here we exploit the phase space formalism to carry out a refined
analysis of the rate of convergence in this quantum central limit theorem. For
instance, we prove that the convergence takes place at a rate
$\mathcal{O}\left(n^{-1/2}\right)$ in the Hilbert--Schmidt norm whenever the
third moments of $\rho$ are finite. Trace norm or relative entropy bounds can
be obtained by leveraging the energy boundedness of the state. Via analytical
and numerical examples we show that our results are tight in many respects. An
extension of our proof techniques to the non-i.i.d. setting is used to analyse
a new model of a lossy optical fibre, where a given $m$-mode state enters a
cascade of $n$ beam splitters of equal transmissivities $\lambda^{1/n}$ fed
with an arbitrary (but fixed) environment state. Assuming that the latter has
finite third moments, and ignoring unitaries, we show that the effective
channel converges in diamond norm to a simple thermal attenuator, with a rate
$\mathcal{O}\Big(n^{-\frac{1}{2(m+1)}}\Big)$. This allows us to establish
bounds on the classical and quantum capacities of the cascade channel. Along
the way, we derive several results that may be of independent interest. For
example, we prove that any quantum characteristic function $\chi_\rho$ is
uniformly bounded by some $\eta_\rho<1$ outside of any neighbourhood of the
origin; also, $\eta_\rho$ can be made to depend only on the energy of the state
$\rho$.
