  Arbitrary quantum states cannot be copied. In fact, to make a copy we must
provide complete information about the system. However, can a quantum system
self-replicate? This is not answered by the no-cloning theorem. In the
classical context, Von Neumann showed that a `universal constructor' can exist
which can self-replicate an arbitrary system, provided that it had access to
instructions for making copy of the system. We question the existence of a
universal constructor that may allow for the self-replication of an arbitrary
quantum system. We prove that there is no deterministic universal quantum
constructor which can operate with finite resources. Further, we delineate
conditions under which such a universal constructor can be designed to operate
dterministically and probabilistically.
