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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., Comment: 17 pages, 9 figures

Past studies of the billiard-ball paradox, a problem involving an object that travels back in time along a closed timelike curve (CTC), typically concern themselves with entirely classical histories, whereby any trajectorial effects associated with quantum mechanics cannot manifest. Here we develop a quantum version of the paradox, wherein a (semiclassical) wave packet evolves through a region containing a wormhole time machine. This is accomplished by mapping all relevant paths on to a quantum circuit, in which the distinction of the various paths is facilitated by representing the billiard particle with a clock state. For this model, we find that Deutsch's prescription (D-CTCs) provides self-consistent solutions in the form of a mixed state composed of terms which represent every possible configuration of the particle's evolution through the circuit. In the equivalent circuit picture (ECP), this reduces to a binomial distribution in the number of loops of time machine. The postselected teleportation prescription (P-CTCs) on the other hand predicts a pure-state solution in which the loop counts have binomial coefficient weights. We then discuss the model in the continuum limit, with a particular focus on the various methods one may employ in order to guarantee convergence in the average number of clock evolutions. Specifically, for D-CTCs, we find that it is necessary to regularise the theory's parameters, while P-CTCs alternatively require more contrived modification., Comment: 18 pages, 9 figures

General relativity predicts the existence of closed timelike curves (CTCs), along which an object could travel to its own past. A consequence of CTCs is the failure of determinism, even for classical systems: one initial condition can result in multiple evolutions. Here we introduce a new quantum formulation of a classic example, where a billiard ball can travel along two possible trajectories: one unperturbed and one, along a CTC, where it collides with its past self. Our model includes a vacuum state, allowing the ball to be present or absent on each trajectory, and a clock, which provides an operational way to distinguish the trajectories. We apply the two foremost quantum theories of CTCs to our model: Deutsch's model (D-CTCs) and postselected teleportation (P-CTCs). We find that D-CTCs reproduce the classical solution multiplicity in the form of a mixed state, while P-CTCs predict an equal superposition of the two trajectories, supporting a conjecture by Friedman et al. [Phys. Rev. D 42, 1915 (1990)]., Comment: 15 pages, 8 figures