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arXiv:gr-qc/9409003 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Nonlinearity in quantum theory and closed timelike curves

Michael J. Cassidy

Published Nov 15, 1995• 19 citations
arXivHTMLPDF

Abstract

We examine the consequences of the density matrix approach to quantum theory in the context of a model spacetime containing closed timelike curves. We find that, in general, an initially pure state will evolve in a nonlinear way to a mixed quantum state. CPT invariance and the implications of this nonlinearity for the statistical interpretation of quantum theory are discussed.

DOI
10.1103/physrevd.52.5676

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We examine the consequences of the density matrix approach to quantum theory in the context of a model spacetime containing closed timelike curves. We find that, in general, an initially pure state will evolve in a nonlinear way to a mixed quantum state. CPT invariance and the implications of this nonlinearity for the statistical interpretation of quantum theory are discussed.
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.
Two formulations of quantum mechanics, inequivalent in the presence of closed timelike curves, are studied in the context of a soluable system. It illustrates how quantum field nonlinearities lead to a breakdown of unitarity, causality, and superposition using a path integral. Deutsch's density matrix approach is causal but typically destroys coherence. For each of these formulations I demonstrate that there are yet further alternatives in prescribing the handling of information flow (inequivalent to previous analyses) that have implications for any system in which unitarity or coherence is not preserved.
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.
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.