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.
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.
Closed timelike curves (CTCs) are a theoretical testbed for how quantum mechanics behaves when causal order breaks down. Two major quantum models exist: Deutsch CTCs, which impose a fixed-point self-consistency condition on the state inside the loop, and postselected CTCs, which model time travel via teleportation and postselection. Although both avoid logical paradoxes, they generally produce different quantum evolutions for the same circuit. This paper compares the two models directly at the circuit level using a small density-matrix and statevector simulator. The Deutsch model is implemented by solving for self-consistent fixed points; the postselected model by applying the corresponding nonlinear postselection map. Identical test circuits, including grandfather-paradox circuits, CNOT, SWAP, controlled-unitary families, and Haar-random unitaries, are run through both models, and the outputs compared using trace distance. The two models are found to disagree generically, not just numerically. The disagreement stems from a structural difference: Deutsch CTCs enforce consistency with mixed fixed-point states, while postselected CTCs renormalize only successful histories. For example, the X-gate "grandfather" circuit yields a well-defined mixed state under the Deutsch rule but an undefined (null-projection) output under postselection. Across 200 random unitaries, the two prescriptions never agree. All simulation code is included for full reproducibility.
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.
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.
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.
The consistency of quantum mechanics in the presence of Closed Timelike Curves (CTCs) has traditionally been framed as a kinematic constraint. We propose a general dynamic framework where causal consistency emerges as a thermodynamic fixed point of open quantum systems. By modeling retrocausal loops as non-linear quantum channels with dissipation, we prove a Universal Contraction Theorem using the logarithmic norm of the Liouvillian superoperator. We demonstrate that any feedback system where the dissipation rate γ exceeds the non-linear Lipschitz constant K must relax to a unique, paradox-free state. We apply this framework to superconducting transmon qubits, deriving quantitative stability bounds (K < 10⁴ s⁻¹) testable with current FPGA technology. This work unifies the arrow of time with the Second Law of Thermodynamics, suggesting that causality is an emergent phase of non-equilibrium matter.
Closed timelike curves (CTCs) are non-intuitive theoretical solutions of general relativity field equations. The main paradox associated with the physical existence of CTCs, the so-called grandfather paradox, can be satisfactorily solved by a quantum model named Deutsch-CTC. An outstanding theoretical result that has been demonstrated in the Deutsch-CTC model is the computational equivalence of a classical and a quantum computer in the presence of a CTC. In this article, in order to explore the possible implications for the foundations of quantum mechanics of that equivalence, a fundamental particle is modelled as a classical-like system supplemented with an information space in which a randomizer and a classical Turing machine are stored. The particle could then generate quantum behavior in real time in case it was controlled by a classical algorithm coding the rules of quantum mechanics and, in addition, a logical circuit simulating a CTC was present on its information space. The conditions that, through the action of evolution under natural selection, might produce a population of such particles with both elements on their information spaces from initial sheer random behavior are analyzed.