We discuss the concept of proper and improper density matrixes and we argue that this issue is of fundamental importance for the understanding of the quantum-mechanical CTC model proposed by Deutsch. We arrive at the conclusion that under a realistic interpretation, the distinction between proper and improper density operators is not a relativistically covariant notion and this fact leads to the conclusion that the D-CTC model is physically inconsistent.
We discuss the concept of proper and improper density matrixes and we argue that this issue is of fundamental importance for the understanding of the quantum-mechanical CTC model proposed by Deutsch. We arrive at the conclusion that under a realistic interpretation, the distinction between proper and improper density operators is not a relativistically covariant notion and this fact leads to the conclusion that the D-CTC model is physically inconsistent.
We discuss the nature of quantum states (density operators) and entanglement in quantum theory with closed timelike curves. One of the key concepts in the standard quantum theory is that given any mixed state we can always purify it in an enlarged Hilbert space by bringing an ancillary system. The purified state does not depend on the state of any extraneous system with which the mixed state is going to interact and on the physical interaction. Here, we prove that it is not possible to purify a mixed state that traverses a closed time like curve (CTC) and allowed to interact in a consistent way with a causality‐respecting (CR) quantum system. Thus, in general for arbitrary interactions between the CR and the CTC systems there is no universal ‘Church of the larger Hilbert space’ for mixed states with CTC. This shows that in quantum theory with CTCs there can exist ‘proper’ and ‘improper’ mixtures. Also, we suggest what kind of entangled states are allowed in the presence of CTCs. We argue that the nature of entanglement can be different in the presence of such exotic objects. Towards the end we discuss how the no‐hiding theorem in the presence of CTCs may allow quantum information pop‐up.
We show that it is possible to clone quantum states to arbitrary accuracy in the presence of a Deutschian closed timelike curve (D-CTC), with a fidelity converging to one in the limit as the dimension of the CTC system becomes large--thus resolving an open conjecture [Brun et al., Phys. Rev. Lett. 102, 210402 (2009)]. This result follows from a D-CTC-assisted scheme for producing perfect clones of a quantum state prepared in a known eigenbasis, and the fact that one can reconstruct an approximation of a quantum state from empirical estimates of the probabilities of an informationally complete measurement. Our results imply more generally that every continuous, but otherwise arbitrarily nonlinear map from states to states, can be implemented to arbitrary accuracy with D-CTCs. Furthermore, our results show that Deutsch's model for closed timelike curves is in fact a classical model, in the sense that two arbitrary, distinct density operators are perfectly distinguishable (in the limit of a large closed timelike curve system); hence, in this model quantum mechanics becomes a classical theory in which each density operator is a distinct point in a classical phase space.
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.
In a recent Letter, Bennett and coworkers [1] argue that proofs of exotic quantum effects using closed timelike curves (CTC's) based on the work of Deutsch [2], or other nonlinear quantum dynamics, suffer from a fallacy that they call the "linearity trap," by which one cannot assume that a classical mixture of input states will lead to the same mixture of corresponding output states. We show that this prescription is inconsistent with the assumption that one can verify the postulated nonlinear evolution. Specifically, if some agent can empirically verify the deterministic nonlinear action of a physical device for some set of pure-state inputs, then one cannot generally use density matrices to represent the input for a second agent who is ignorant of the actual pure state prepared. Our argument is epistemological and makes no appeal to the detailed physics of CTC's.
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.
How should we think about subjective states vs. objective states? Is it a question of the meaning associated with a state? Recent work on this question arose in consideration of closed timelike curves (CTCs) and their possible role in quantum computers. The issue of ontic and epistemic states is particularly important when considering CTCs because, as one may argue, the interpretation of quantum states as either ontic or epistemic will naturally lead to different assumptions about how quantum systems behave in the presence of CTCs. For example, David Deutsch studied various time travel scenarios in a classical model and then in a quantum model motivated by a strictly ontic interpretation of quantum states. While in the classical model, CTCs could produce paradoxes, however Deutsch argued that no paradoxes can occur in his parallel universes (modified Everett interpretation) quantum treatment. Although all paradoxes are resolved in this way, the resulting theory is not standard quantum theory, but a new nonlinear theory. Many implications can arise using Deutsch's model and I shall discuss some of them in particular. These assumptions are particularly unconventional in part because they require that mixed quantum states are ontic. Pure quantum states can be interpreted as ontic, however most interpretations view mixed states as epistemic, i.e., reflecting an observer's lack of knowledge. I shall here consider an alternative proposal-pure quantum states, represented by density matrices containing off-diagonal elements, are possibly epistemic (since we never actually see them) while mixed quantum states arising in CTCs are always ontic representing the action of consciousness in observers (we do see them). Paradoxically my argument is based on the well-known experiences of gaining knowledge in classical physics; couching this in quantum physics terms, classical mixed states (represented by diagonal density matrices in quantum physics) are just what arise when an "observation" is said to occur resulting in a so-called reduction of the quantum wave function and the appearance of a classical world. In brief, Deutsch's CTC nonlinear post quantum physics model may represent the action of a conscious mind.