I recently came across the paper Non Existence of Quantum Mechanical Self Replicating Machine by Chakrabarty and Indranil which builds on Quantum mechanical Universal constructor by Patil and Braunstein.
The paper shows that it is not possible to make a self-replicating quantum mechanical device because of the linearity of quantum mechanics, the no cloning theorems and the conservation of entanglement. The unitariness of quantum mechanics is not very favorable towards replication: a real replicator needs to convert arbitrary states into states corresponding to a copy, but quantum mechanics is information preserving. We need decoherence to replicate. Presumably one could construct an approximative replicator that does not achieve perfect self-replication but only “good enough” replication (the Patil paper shows that one can make a replicator that produces a finite number of generations, and mentions the possibility of indeterministic replicators).
Via Anders Sandberg.