  Any given prescription of quantum time travel necessarily endows a Hilbert
space to the chronology-violating (CV) system on the closed timelike curve
(CTC). However, under the two foremost models, Deutsch's prescription (D-CTCs)
and postselected teleportation (P-CTCs), the CV system is treated very
differently: D-CTCs assign a definite form to the state on this system, while
P-CTCs do not. To further explore this distinction, we present a methodology by
which an operational notion of state may be assigned to their respective CV
systems. This is accomplished via a conjunction of state tomography and weak
measurements, with the latter being essential in leaving any notions of
self-consistency intact. With this technique, we are able to verify the
predictions of D-CTCs and, perhaps more significantly, operationally assign a
state to the system on the P-CTC. We show that, for any given combination of
chronology-respecting input and unitary interaction, it is always possible to
recover the unique state on the P-CTC, and we provide a few specific examples
in the context of select archetypal temporal paradoxes. We also demonstrate how
this state may be derived from analysis of the P-CTC prescription itself, and
we explore how it compares to its counterpart in the CV state predicted by
D-CTCs.
