  Entanglement distillation transforms weakly entangled noisy states into
highly entangled states, a primitive to be used in quantum repeater schemes and
other protocols designed for quantum communication and key distribution. In
this work, we present a comprehensive framework for continuous-variable
entanglement distillation schemes that convert noisy non-Gaussian states into
Gaussian ones in many iterations of the protocol. Instances of these protocols
include (a) the recursive-Gaussifier protocol, (b) the temporally-reordered
recursive-Gaussifier protocol, and (c) the pumping-Gaussifier protocol. The
flexibility of these protocols give rise to several beneficial trade-offs
related to success probabilities or memory requirements, which that can be
adjusted to reflect experimental demands. Despite these protocols involving
measurements, we relate the convergence in this protocols to new instances of
non-commutative central limit theorems, in a formalism that we lay out in great
detail. Implications of the findings for quantum repeater schemes are
discussed.
