Abstract This study examines closed timelike curves (CTCs) in quantum mechanics, focusing on the Deutsch model, one of the most widely studied frameworks for describing quantum systems in the presence of time loops. In this model, a quantum state ∣ψ⟩|\psi\rangle∣ψ⟩ interacts unitarily with its past version, effectively forming a two-qubit system where the chronologically consistent qubit interacts with a qubit ρCTC\rho_{\text{CTC}}ρCTC trapped in a closed timelike curve. The state ρCTC\rho_{\text{CTC}}ρCTC is determined by the Deutsch consistency condition, ensuring self-consistent evolution along the CTC. This approach allows for rigorous modeling of quantum systems under non-classical temporal structures, providing insights into temporal correlations, causality violations, and the potential computational advantages offered by CTC-assisted quantum operations. While alternative CTC models based on postselection have been proposed, the Deutsch model remains a foundational reference for studying quantum evolution in non-linear temporal frameworks. Its analysis informs research in quantum information, retrocausality, and the theoretical limits of computation in exotic spacetime geometries.
Abstract This study examines closed timelike curves (CTCs) in quantum mechanics, focusing on the Deutsch model, one of the most widely studied frameworks for describing quantum systems in the presence of time loops. In this model, a quantum state ∣ψ⟩|\psi\rangle∣ψ⟩ interacts unitarily with its past version, effectively forming a two-qubit system where the chronologically consistent qubit interacts with a qubit ρCTC\rho_{\text{CTC}}ρCTC trapped in a closed timelike curve. The state ρCTC\rho_{\text{CTC}}ρCTC is determined by the Deutsch consistency condition, ensuring self-consistent evolution along the CTC. This approach allows for rigorous modeling of quantum systems under non-classical temporal structures, providing insights into temporal correlations, causality violations, and the potential computational advantages offered by CTC-assisted quantum operations. While alternative CTC models based on postselection have been proposed, the Deutsch model remains a foundational reference for studying quantum evolution in non-linear temporal frameworks. Its analysis informs research in quantum information, retrocausality, and the theoretical limits of computation in exotic spacetime geometries.
Although general relativity allows the existence of closed timelike curves (CTCs), self-consistency problems arise (the ‘grandfather paradox’ among others). It is known that quantum mechanical consideration of the matter formally removes all the paradoxes, but the questions about causal structure remain. On the other hand, the idea of postselected CTCs (P-CTC) in quantum teleportation has know been put forward and experimentally implemented. We consider these problems with the aid of quantum causal analysis, where causality is defined without invoking the time relation. It implements the Cramer principle of weak causality, which admits time reversal in entangled states. We analyze Deutsch CTCs (D-CTC) with different kinds of interactions between the chronology-violating and chronology-respecting particles, with refined inferences about mixedness, quantum/classical correlations, entanglement and thermodynamics in the D-CTC. The main result is that time reversal causality can really exist, however, the final quantum state does not place retrospective constraints on the initial state, instead the final state can influence the state inside the D-CTC. This is effectively the implementation of Novikov self-consistency principle. The P-CTC has radically different properties; in particular, if the initial state was pure, the final state is always pure too. Self-consistency is controlled by the initial state-dependent traversability of the P-CTC.
One out of many emerging implications from solutions of Einstein's general relativity equations are closed timelike curves (CTCs), which are trajectories through space-time that allow for time travel to the past without exceeding the speed of light. Two main quantum models of computation with the use of CTCs were introduced by Deutsch (D-CTC) and by Bennett and Schumacher (P-CTC). Unlike the classical theory in which CTCs lead to logical paradoxes, the quantum D-CTC model provides a solution that is logically consistent due to the self-consistency condition imposed on the evolving system, whereas the quantum P-CTC model chooses such a solution through postselection. Both models are nonequivalent and imply nonstandard phenomena in the field of quantum computation and quantum mechanics. In this paper, we study the implications of these two models on the second law of thermodynamics---the fundamental principle which states that in an isolated system the entropy never decreases. In particular, we construct CTC-based quantum circuits which lead to a decrease in entropy.
There is now a significant body of results on quantum interactions with closed timelike curves (CTCs) in the quantum information literature, for both the Deutsch model of CTC interactions (D-CTCs) and the projective model (P-CTCs). As a consequence, there is a prima facie argument exploiting entanglement that CTC interactions would enable superluminal and, indeed, effectively instantaneous signaling. In cases of spacelike separation between the sender of a signal and the receiver, whether a receiver measures the local part of an entangled state or a disentangled state to access the signal can depend on the reference frame. We propose a consistency condition that gives priority to either an entangled perspective or a disentangled perspective in spacelike-separated scenarios. For D-CTC interactions, the consistency condition gives priority to frames of reference in which the state is disentangled, while for P-CTC interactions the condition selects the entangled state. Using the consistency condition, we show that there is a procedure that allows Alice to signal to Bob in the past via relayed superluminal communications between spacelike-separated Alice and Clio, and spacelike-separated Clio and Bob. This opens the door to time travel paradoxes in the classical domain. Ralph [T. C. Ralph, arXiv:1107.4675 [quant-ph].] first pointed this out for P-CTCs, but we show that Ralph's procedure for a ``radio to the past'' is flawed. Since both D-CTCs and P-CTCs allow classical information to be sent around a spacetime loop, it follows from a result by Aaronson and Watrous [S. Aaronson and J. Watrous, Proc. R. Soc. A 465, 631 (2009)] for CTC-enhanced classical computation that a quantum computer with access to P-CTCs would have the power of PSPACE, equivalent to a D-CTC-enhanced quantum computer.
Abstract 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.
Closed Timelike Curves (CTCs) are intriguing relativistic objects that allow for time travel to the past and can be used as computational resources. In Deutschian Closed Timelike Curves (D-CTCs), due to the monogamy of entanglement, non-local correlations between entangled states are destroyed. In contrast, for Postselected Closed Timelike Curves (P-CTCs), a second variant of CTCs, the non-local correlations are preserved. P-CTCs can be harnessed for the signaling of non-orthogonal states to the past without a disruption of causality. In this paper, we take up signaling to the past and show a method of sending four non-orthogonal states to the past using P-CTCs. After constructing our signaling protocol, we study the causality violations that our protocol results in and put forward two consistency relations to prevent them.
Deutsch closed timelike curves (D-CTCs) are described by a fixed-point condition for a chronology-violating register. We study a finite-dimensional circuit model that places a Hayden--Preskill/Yoshida--Kitaev recovery map inside such a consistency loop. A register-routing construction makes the Deutsch map explicit: an initial SWAP moves the incoming CTC state to an idle dump register, the scrambler and decoder act on the remaining active registers, and a final SWAP writes the recovered message back to the CTC register. When the active branch recovers the message, the induced map on the CTC register is the replacement channel σ↦ρM, with the unique fixed point ρM. We implement the associated Lloyd-type post-selected decoder circuits on quantum hardware and formulate a classical-feedback iteration for the experimentally estimated map. Qiskit simulations and IBM-hardware data for single-qubit instances quantify decoder fidelity, post-selection overhead, routing-dependent noise, and quantum-geometric susceptibility.