  Entanglement distillation refers to the task of transforming a collection of
weakly entangled pairs into fewer highly entangled ones. It is a core
ingredient in quantum repeater protocols, needed to transmit entanglement over
arbitrary distances in order to realise quantum key distribution schemes.
Usually, it is assumed that the initial entangled pairs are i.i.d. distributed
and uncorrelated with each other, an assumption that might not be reasonable at
all in any entanglement generation process involving memory channels. Here, we
introduce a framework that captures entanglement distillation in the presence
of natural correlations arising from memory channels. Conceptually, we bring
together ideas from condensed-matter physics - that of renormalisation and of
matrix-product states and operators - with those of local entanglement
manipulation, Markov chain mixing, and quantum error correction. We identify
meaningful parameter regions for which we prove convergence to maximally
entangled states, arising as the fixed points of a matrix-product operator
renormalisation flow.
