  Closed timelike curves (CTCs) are trajectories in spacetime that effectively
travel backwards in time: a test particle following a CTC can in principle
interact with its former self in the past. CTCs appear in many solutions of
Einstein's field equations and any future quantum version of general relativity
will have to reconcile them with the requirements of quantum mechanics and of
quantum field theory. A widely accepted quantum theory of CTCs was proposed by
Deutsch. Here we explore an alternative quantum formulation of CTCs and show
that it is physically inequivalent to Deutsch's. Because it is based on
combining quantum teleportation with post-selection, the
predictions/retrodictions of our theory are experimentally testable: we report
the results of an experiment demonstrating our theory's resolution of the
well-known `grandfather paradox.
