We use techniques of quantum information theory to analyze the quantum causal histories approach to quantum gravity. While it is consistent to introduce closed timelike curves (CTCs), they cannot generically carry independent degrees of freedom. Moreover, if the effective dynamics of the chronology-respecting part of the system is linear, it should be completely decoupled from the CTCs. In the absence of a CTC, not all causal structures admit the introduction of quantum mechanics. It is possible for those and only those causal structures that can be represented as quantum computational networks. Dynamics of the subsystems should not be unitary or even completely positive. However, we show that other commonly made assumptions ensure the complete positivity of the reduced dynamics.
We use techniques of quantum information theory to analyze the quantum causal histories approach to quantum gravity. While it is consistent to introduce closed timelike curves (CTCs), they cannot generically carry independent degrees of freedom. Moreover, if the effective dynamics of the chronology-respecting part of the system is linear, it should be completely decoupled from the CTCs. In the absence of a CTC, not all causal structures admit the introduction of quantum mechanics. It is possible for those and only those causal structures that can be represented as quantum computational networks. Dynamics of the subsystems should not be unitary or even completely positive. However, we show that other commonly made assumptions ensure the complete positivity of the reduced dynamics.
We provide a precise definition and analysis of quantum causal histories (QCHs). A QCH consists of a discrete, locally finite, causal pre-spacetime with matrix algebras encoding the quantum structure at each event. The evolution of quantum states and observables is described by completely positive maps between the algebras at causally related events. We show that this local description of evolution is sufficient and that unitary evolution can be recovered wherever it should actually be expected. This formalism may describe a quantum cosmology without an assumption of global hyperbolicity; it is thus more general than the Wheeler–De Witt approach. The structure of a QCH is also closely related to quantum information theory and algebraic quantum field theory on a causal set.
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.
Abstract In this paper, we construct a tensor network representation of quantum causal histories, as a step towards directly representing states in quantum gravity via bulk tensor networks. Quantum causal histories are quantum extensions of causal sets in the sense that on each event in a causal set is assigned a Hilbert space of quantum states, and the local causal evolutions between events are modeled by completely positive and trace-preserving maps. Here we utilize the channel-state duality of completely positive and trace-preserving maps to transform the causal evolutions to bipartite entangled states. We construct the matrix product state for a single quantum causal history by projecting the obtained bipartite states onto the physical states on the events. We also construct the two dimensional tensor network states for entangled quantum causal histories in a restricted case with compatible causal orders. The possible holographic tensor networks are explored by mapping the quantum causal histories in a way analogous to the exact holographic mapping. The constructed tensor networks for quantum causal histories are exemplified by the non-unitary local time evolution moves in a quantum system on temporally varying discretizations, and these non-unitary evolution moves are shown to be necessary for defining a bulk causal structure and a quantum black hole. Finally, we comment on the limitations of the constructed tensor networks, and discuss some directions for further studies aiming at applications in quantum gravity.
We propose that quantum entanglement, Einstein–Rosen bridges, and post-selected closed timelike curves are interconnected manifestations of a unified causal-consistency mechanism in quantum gravity. In this framework—Paradox-Constraining Time-Causal Structures (PCTCS)—entanglement supplies retrocausal boundary constraints, ER bridges provide geometric connectivity, and P-CTCs eliminate paradoxical histories via post-selection. This model preserves causality without requiring faster-than-light signaling and offers testable predictions in entangled photon experiments and quantum-computer simulations.
The problem of time in quantum gravity is often presented as a consequence of applying quantum theory to general relativity. These notes adopt a broader perspective. We first examine how time enters classical mechanics, relativity, quantum theory, and quantum field theory, distinguishing three principal aspects: causal order, temporal duration, and the present. This analysis shows that the conceptual difficulties associated with time do not arise only when gravity is quantised. Tensions between causal structure, clocks, observables, and measurement are already present in the theories from which quantum gravity is constructed. We then survey responses to the problem of time across the principal approaches to quantum gravity. These include canonical and Wheeler-DeWitt quantisation, path-integral and histories formulations, perturbative and background-dependent approaches, causal sets, and twistor theory. The programmes are compared according to whether they retain an external time, render time problematic through quantisation, or incorporate causal or temporal structure into their foundations. Particular emphasis is placed on questions arising from quantum foundations and relativistic quantum information. We conclude by examining linearised gravity and weak-gravity quantum systems. This regime shows that the problem of time is not confined to Planck-scale physics, but arises already in attempts to provide a predictive and operationally meaningful quantum description of spacetime near flat geometry.