This thesis studies fixed point non-uniqueness in Deutsch's model of time travel along a closed timelike curve (CTC), primarily focusing upon the bootstrap problem in the two-qubit setting. This thesis focuses on how the under-determination stemming from the bootstrap problem is structured, how prevalent it is, and whether it can be resolved by a physical mechanism, rather than an externally imposed rule. These questions are primarily addressed in the restricted setting of a two-qubit system, where mathematical criteria for uniqueness and non-uniqueness are developed analytically. From these criteria, it is shown that in the two-qubit setting, the non-unique sector is confined to a thin exceptional subset of the full parameter space. The thesis then turns to examine noise as a physical mechanism for resolving under-determination, and analytically characterizes the types of channels that will universally select a unique fixed point when composed with single-qubit D-CTC systems.
This thesis studies fixed point non-uniqueness in Deutsch's model of time travel along a closed timelike curve (CTC), primarily focusing upon the bootstrap problem in the two-qubit setting. This thesis focuses on how the under-determination stemming from the bootstrap problem is structured, how prevalent it is, and whether it can be resolved by a physical mechanism, rather than an externally imposed rule. These questions are primarily addressed in the restricted setting of a two-qubit system, where mathematical criteria for uniqueness and non-uniqueness are developed analytically. From these criteria, it is shown that in the two-qubit setting, the non-unique sector is confined to a thin exceptional subset of the full parameter space. The thesis then turns to examine noise as a physical mechanism for resolving under-determination, and analytically characterizes the types of channels that will universally select a unique fixed point when composed with single-qubit D-CTC systems.
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.
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.
This preprint investigates environmental effects in the Deutsch model of quantum closed timelike curves (CTCs). In the Deutsch framework, consistency of a time loop is enforced by a fixed-point condition on the quantum state circulating within the loop, leading to effective nonlinear dynamics on interacting systems. Focusing on finite-dimensional Deutsch CTC models with well-defined fixed-point selection rules, we examine whether loop self-consistency alone implies environmental invariance, meaning that degrees of freedom external to the loop remain unchanged after a traversal. Using explicit system–environment couplings, we analyze the reduced environment dynamics induced by a single loop interaction. We show that for generic nontrivial couplings, there exist environment preparations for which a detectable disturbance is induced, even though the loop state satisfies the Deutsch self-consistency condition. Exact environmental invariance arises only in the trivial, factorized coupling limit within the class of models considered. These results distinguish internal paradox resolution on the loop register from the operational behavior of external degrees of freedom, indicating that environmental invariance is not enforced by Deutsch self-consistency alone. The conclusions are independent of the specific fixed-point selection rule and rely only on the structural properties of Deutsch consistency and completely positive dynamics. The analysis provides an operational framework for comparing environmental signatures of different quantum CTC constructions and can be extended to alternative consistency models, including postselected and trace-preserving formulations.
We study the question of what is computable by Turing machines equipped with time travel into the past; i.e., with Deutschian closed timelike curves (CTCs) having no bound on their width or length. An alternative viewpoint is that we study the complexity of finding approximate fixed points of computable Markov chains and quantum channels of countably infinite dimension. Our main result is that the complexity of these problems is precisely Δ2, the class of languages Turing-reducible to the Halting problem. Establishing this as an upper bound for qubit-carrying CTCs requires recently developed results in the theory of quantum Markov maps.
This dissertation is an exploration of several issues surrounding David Deutsch’s CTC model first introduced in his 1991 paper “Quantum Mechanics Near Closed Timelike Lines”. Deutsch developed his model to account for the effects of quantum theory, which had been left out of classical discussions of time travel paradoxes. Deutsch’s formulation of his model in terms of quantum computational circuits lends itself to being adopted in the quantum information community. The dissertation argues that the adoption of the D-CTC model entails the existence of Nonlocal Signaling, which is in conflict with a fundamental principle of the quantum information approach. In order to motivate this argument, in Chapter 2 I introduce a distinction between Nonlocal Signaling, and Superluminal Information Transfer. In the latter case, a carrier of information physically traverses the space between the distant communicating parties faster than the speed of light. Exploiting quantum entanglement to signal, however, need not have this feature. I term this Nonlocal Signaling. Chapter 3 is where I present the argument that D-CTCs entail Nonlocal Signaling, and examine the controversy surrounding this and related results. I argue that the resistance to these kinds of predictions in the literature is motivated by a commitment to the principles of quantum information theory, which are inappropriately applied here. Chapters 4 and 5 examine details of Deutsch’s model. Chapter 4 argues that it presupposes a significant metaphysical picture that, when explicitly stated, makes a much less comfortable fit between D-CTCs and quantum information theory. Chapter 5 argues that, because of Deutsch’s commitment to this metaphysical picture, he is committed to the existence of physical situations that are in every way indistinguishable from the paradoxes he attempts to rule out by adopting the model in the first place. In Chapter 6, I make some observations about the relationship between the quantum information-theoretic approach to the interpretation of quantum theory, and the approaches focused primarily on arguing for one or another underlying ontology. Deutsch’s model is situated squarely in the latter camp. It serves as a useful example in pulling apart the implications of the two approaches. In conclusion, I argue that the quantum information-theoretic interpretation of quantum theory, in denying the fundamentality of any particular ontology, in favor of kinematical principles, is in tension with the metaphysical commitments of the Deutsch model. Deutsch’s interpretational stance is among the metaphysically-motivated positions. I argue that this element of the Deutsch model is essential to the solutions it offers to the paradoxes of time travel, and therefore the D-CTC model cannot be adopted without implicitly endorsing Deutsch’s metaphysical commitments. This feature makes the D-CTC model an uncomfortable fit with QIT.
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.