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arXiv:1003.1987 · Physical Review A

Information flow of quantum states interacting with closed timelike curves

Timothy C. Ralph, Casey R. Myers

Published Dec 29, 2010• 37 citations
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Abstract

Recently, the quantum information processing power of closed timelike curves has been discussed. Because the most widely accepted model for quantum closed timelike curve interactions contains ambiguities, different authors have been able to reach radically different conclusions as to the power of such interactions. By tracing the information flow through such systems we are able to derive equivalent circuits with unique solutions, thus allowing an objective decision between the alternatives to be made. We conclude that closed timelike curves, if they exist and are well described by these simple models, would be a powerful resource for quantum information processing.

DOI
10.1103/physreva.82.062330

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Comment on “Information flow of quantum states interacting with closed timelike curves”

Waldemar Kłobus, Andrzej Grudka, Antoni Wójcik

We show that recent results on the interaction of causality-respecting particles with particles on closed timelike curves derived in [Phys. Rev. A 82, 062330 (2010)] depend on ambiguous assumptions about the form of the state which is inputted into the proposed equivalent circuit. Choosing a different form of this state leads to an opposite conclusion on the power of closed timelike curves.

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Reply to “Comment on ‘Information flow of quantum states interacting with closed timelike curves’”

Timothy C. Ralph, Casey R. Myers

We respond to the comment by K\l{}obus et al. [Phys. Rev. A 84, 056301 (2011)] by emphasizing that the equivalent circuit, once constructed, obeys the standard rules of quantum mechanics---hence there is no ambiguity in how to choose initial states in our model. We discuss the distinction between correlated ensembles produced nonlocally via measurements on entangled states and those produced via local preparation.

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Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which represents a valid quantification of resources given the ability to construct compact regions of closed timelike curves. The notion of self-consistent evolution for quantum computers whose components follow closed timelike curves, as pointed out by Deutsch [Phys. Rev. D 44, 3197 (1991)], implies that the evolution of the chronology respecting components which interact with the closed timelike curve components is nonlinear. We demonstrate that this nonlinearity can be used to efficiently solve computational problems which are generally thought to be intractable. In particular we demonstrate that a quantum computer which has access to closed timelike curve qubits can solve $\mathit{NP}$-complete problems with only a polynomial number of quantum gates.

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In orthodox quantum theory the observables of spacelike separated quantum systems commute. I shall call this the commutation constraint. It severely limits quantum theory's explanatory power. For instance, the constraint cannot be met in the presence of closed timelike curves, leaving us with no choice but to rule them out by fiat. It also conflicts with Bekenstein's bound. Here I investigate a modified quantum theory, unorthodox quantum theory, which is different from the conventional theory only in its omission of this commutation constraint. In particular, I describe a system of unorthodox qubits and demonstrate how they can be used to model systems on closed timelike curves and how they allow for a solution of the grandfather paradox.

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Recently, the quantum information processing power of closed timelike curves has been discussed. Because the most widely accepted model for quantum closed timelike curve interactions contains ambiguities, different authors have been able to reach radically different conclusions as to the power of such interactions. By tracing the information flow through such systems we are able to derive equivalent circuits with unique solutions, thus allowing an objective decision between the alternatives to be made. We conclude that closed timelike curves, if they exist and are well described by these simple models, would be a powerful resource for quantum information processing.
While closed timelike curves (CTCs) are not known to exist, studying their consequences has led to non-trivial insights into general relativity, quantum information and other areas. In this paper, we show that, if CTCs existed, quantum computers would be no more powerful than classical computers: both would have the (extremely large) power of the complexity class polynomial space ( ), consisting of all problems solvable by a conventional computer using a polynomial amount of memory. This solves an open problem proposed by one of us in 2005, and gives an essentially complete understanding of computational complexity in the presence of CTCs. Following the work of Deutsch, we treat a CTC as simply a region of spacetime where a ‘causal consistency’ condition is imposed, meaning that nature has to produce a (probabilistic or quantum) fixed point of some evolution operator. Our conclusion is then a consequence of the following theorem: given any quantum circuit (not necessarily unitary), a fixed point of the circuit can be (implicitly) computed in . This theorem might have independent applications in quantum information.
Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which represents a valid quantification of resources given the ability to construct compact regions of closed timelike curves. The notion of self-consistent evolution for quantum computers whose components follow closed timelike curves, as pointed out by Deutsch [Phys. Rev. D 44, 3197 (1991)], implies that the evolution of the chronology respecting components which interact with the closed timelike curve components is nonlinear. We demonstrate that this nonlinearity can be used to efficiently solve computational problems which are generally thought to be intractable. In particular we demonstrate that a quantum computer which has access to closed timelike curve qubits can solve NP\mathit{NP}NP-complete problems with only a polynomial number of quantum gates.
In orthodox quantum theory the observables of spacelike separated quantum systems commute. I shall call this the commutation constraint. It severely limits quantum theory's explanatory power. For instance, the constraint cannot be met in the presence of closed timelike curves, leaving us with no choice but to rule them out by fiat. It also conflicts with Bekenstein's bound. Here I investigate a modified quantum theory, unorthodox quantum theory, which is different from the conventional theory only in its omission of this commutation constraint. In particular, I describe a system of unorthodox qubits and demonstrate how they can be used to model systems on closed timelike curves and how they allow for a solution of the grandfather paradox.
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.