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arXiv:2205.02797

On unorthodox qubits, with an application to the closed timelike curve problem

Samuel Kuypers

Published May 5, 2022• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2205.02797

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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.
Closed timelike curves (CTCs) are trajectories in spacetime that effectively travel backwards in time: a test particle following a CTC can interact with its former self in the past. A widely accepted quantum theory of CTCs was proposed by Deutsch. Here we analyze an alternative quantum formulation of CTCs based on teleportation and postselection, and show that it is inequivalent to Deutsch's. The predictions or retrodictions of our theory can be simulated experimentally: we report the results of an experiment illustrating how in our particular theory the "grandfather paradox" is resolved.
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.
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.
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.
The methods of the quantum theory of computation are used to analyze the physics of closed timelike lines. This is dominated, even at the macroscopic level, by quantum mechanics. In classical physics the existence of such lines in a spacetime imposes "paradoxical" constraints on the state of matter in their past and also provides means for knowledge to be created in ways that conflict with the principles of the philosophy of science. In quantum mechanics the first of these pathologies does not occur. The second is mitigated, and may be avoidable without such spacetimes being ruled out. Several novel and distinctive (but nonparadoxical) quantum-mechanical effects occur on and near closed timelike lines, including violations of the correspondence principle and of unitarity. It becomes possible to "clone" quantum systems and to measure the state of a quantum system. A new experimental test of the Everett interpretation against all others becomes possible. Consideration of these and other effects sheds light on the nature of quantum mechanics.
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.