  Toy models for quantum evolution in the presence of closed timelike curves
(CTCs) have gained attention in the recent literature due to the strange
effects they predict. The circuits that give rise to these effects appear quite
abstract and contrived, as they require non-trivial interactions between the
future and past which lead to infinitely recursive equations. We consider the
special case in which there is no interaction inside the CTC, referred to as an
open timelike curve (OTC), for which the only local effect is to increase the
time elapsed by a clock carried by the system. Remarkably, circuits with access
to OTCs are shown to violate Heisenberg's uncertainty principle, allowing
perfect state discrimination and perfect cloning of coherent states. The model
is extended to wave-packets and smoothly recovers standard quantum mechanics in
an appropriate physical limit. The analogy with general relativistic
time-dilation suggests that OTCs provide a novel alternative to existing
proposals for the behaviour of quantum systems under gravity.
