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arXiv:2102.03408 · Physical review. D/Physical review. D.

Relativistic causality in particle detector models: Faster-than-light signaling and impossible measurements

José de Ramón, Maria Papageorgiou, Eduardo Martín-Martínez

Published Apr 9, 2021• 46 citations
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Abstract

We analyze potential violations of causality in Unruh DeWitt-type detector models in relativistic quantum information. We proceed by first studying the relation between faster-than-light signaling and the causal factorization of the dynamics for multiple detector-field interactions. We show in what way spatially extended nonrelativistic detector models predict superluminal propagation of the field's initial conditions. We draw parallels between this characteristic of detector models, stemming from their nonrelativistic dynamics, and Sorkin's ``impossible measurements on quantum fields'' [R. D. Sorkin, in Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland (Cambridge University Press, 1993), Vol. 2, pp. 293--305.]. Based on these features, we discuss the validity of measurements in quantum field theory when performed with nonrelativistic particle detectors.

DOI
10.1103/physrevd.103.085002

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We analyze potential violations of causality in Unruh DeWitt-type detector models in relativistic quantum information. We proceed by first studying the relation between faster-than-light signaling and the causal factorization of the dynamics for multiple detector-field interactions. We show in what way spatially extended nonrelativistic detector models predict superluminal propagation of the field's initial conditions. We draw parallels between this characteristic of detector models, stemming from their nonrelativistic dynamics, and Sorkin's ``impossible measurements on quantum fields'' [R. D. Sorkin, in Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland (Cambridge University Press, 1993), Vol. 2, pp. 293--305.]. Based on these features, we discuss the validity of measurements in quantum field theory when performed with nonrelativistic particle detectors.
It is shown that an ideal measurement of a one–particle wave packet state of a relativistic quantum field in Minkowski spacetime enables superluminal signalling. The result holds for a measurement that takes place over an intervention region in spacetime whose extent in time in some frame is longer than the light crossing time of the packet in that frame. Moreover, these results are shown to apply not only to ideal measurements but also to unitary transformations that rotate two orthogonal one–particle states into each other. In light of these observations, possible restrictions on the allowed types of intervention are considered. A more physical approach to such questions is to construct explicit models of the interventions as interactions between the field and other quantum systems such as detectors. The prototypical Unruh-DeWitt detector couples to the field operator itself and so most likely respects relativistic causality. On the other hand, detector models which couple to a finite set of frequencies of field modes are shown to lead to superluminal signalling. Such detectors do, however, provide successful phenomenological models of atom-qubits interacting with quantum fields in a cavity but are valid only on time scales many orders of magnitude larger than the light crossing time of the cavity.
It is shown that an ideal measurement of a one-particle wave packet state of a relativistic quantum field in Minkowski spacetime enables superluminal signalling. The result holds for a measurement that takes place over an intervention region in spacetime whose extent in time in some frame is longer than the light-crossing time of the packet in that frame. Moreover, these results are shown to apply not only to ideal measurements but also to unitary transformations that rotate two orthogonal one-particle states into each other. In light of these observations, possible restrictions on the allowed types of intervention are considered. A more physical approach to such questions is to construct explicit models of the interventions as interactions between the field and other quantum systems such as detectors. The prototypical Unruh–DeWitt detector couples to the field operator itself and so most likely respects relativistic causality. On the other hand, detector models which couple to a finite set of frequencies of field modes are shown to lead to superluminal signalling. Such detectors do, however, provide successful phenomenological models of atom–qubits interacting with quantum fields in a cavity but are valid only on time scales many orders of magnitude larger than the light-crossing time of the cavity.
In this paper we analyze the problem of ``apparent'' superluminal signalling and retrocausation that can appear for particle detector models when considering noncompactly supported field-detector interactions in quantum field theory in curved spacetimes and in relativistic quantum information protocols. For this purpose, we define a signalling estimator based on an adapted version of the quantum Fisher information to perturbative regimes. This allows us to study how the internal dynamics of the detectors (for example the gap between the detector energy levels) have an impact on the ability of particle detectors to communicate with one another. Moreover, we show that, very generally, even for detectors with infinite tails in space and time, if the tails decay exponentially, one can define an effective light cone, outside of which signalling is negligible. This provides concrete evidence supporting the use of noncompact (but exponentially decaying) detector smearings in protocols of relativistic quantum information.
We propose a measurement theory for quantum fields based on measurements made with localized nonrelativistic systems that couple covariantly to quantum fields (like the Unruh-DeWitt detector). Concretely, we analyze the positive operator-valued measure induced on the field when an idealized measurement is carried out on the detector after it coupled to the field. Using an information-theoretic approach, we provide a relativistic analogue to the quantum-mechanical L\"uders update rule to update the field state following the measurement on the detector. We argue that this proposal has all the desirable characteristics of a proper measurement theory. In particular it does not suffer from the ``impossible measurements'' problem pointed out by Rafael Sorkin in the 1990s which shows that idealized measurements cannot be used in quantum field theory.
We show that the predictions of commonly used spatially smeared particle detectors coupled to quantum fields are not generally covariant outside the pointlike limit. This lack of covariance manifests itself as an ambiguity in the time-ordering operation. We analyze how the breakdown of covariance affects typical detector models in quantum field theory such as the Unruh--DeWitt model. Specifically, we show how the violations of covariance depend on the state of the detectors-field system, the shape and state of motion of the detectors, and the spacetime geometry. Furthermore, we provide the tools to explicitly evaluate the magnitude of the violation and identify the regimes where the predictions of smeared detectors are either exactly or approximately covariant in perturbative analyses, thus providing limits of validity of smeared particle detector models.
We analyze the constraints that causality imposes on some of the particle detector models employed in quantum field theory in general and, in particular, on those used in quantum optics (or superconducting circuits) to model atoms interacting with light. Namely, we show that disallowing faster-than-light communication can impose severe constraints on the applicability of particle detector models in three different common scenarios: (1) when the detectors are spatially smeared, (2) when a UV cutoff is introduced in the theory and (3) under one of the most typical approximations made in quantum optics: the rotating-wave approximation. We identify the scenarios in which the models' causal behavior can and cannot be cured.