  We show that it is possible to clone quantum states to arbitrary accuracy in
the presence of a Deutschian closed timelike curve (D-CTC), with a fidelity
converging to one in the limit as the dimension of the CTC system becomes
large---thus resolving an open conjecture from [Brun et al., Physical Review
Letters 102, 210402 (2009)]. This result follows from a D-CTC-assisted scheme
for producing perfect clones of a quantum state prepared in a known eigenbasis,
and the fact that one can reconstruct an approximation of a quantum state from
empirical estimates of the probabilities of an informationally-complete
measurement. Our results imply more generally that every continuous, but
otherwise arbitrarily non-linear map from states to states can be implemented
to arbitrary accuracy with D-CTCs. Furthermore, our results show that Deutsch's
model for CTCs is in fact a classical model, in the sense that two arbitrary,
distinct density operators are perfectly distinguishable (in the limit of a
large CTC system); hence, in this model quantum mechanics becomes a classical
theory in which each density operator is a distinct point in a classical phase
space.
