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arXiv:2301.08775 · Journal of High Energy Physics

Entanglement structure of quantum fields through local probes

Bruno de S. L. Torres, Kelly Wurtz, José Polo-Gómez, Eduardo Martín-Martínez

Published May 8, 2023• 17 citations
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Abstract

Abstract We present a framework to study the entanglement structure of a quantum field theory inspired by the formalism of particle detectors in relativistic quantum information. This framework can in principle be used to faithfully capture entanglement in a QFT between arbitrary-shaped regions of spacetime without encountering UV divergences, bypassing many of the issues typically present in other approaches. Our results also establish the limits of the efficiency of entanglement harvesting, and may also be used to motivate an operational definition of entanglement between spacetime subregions in field theory.

DOI
10.1007/jhep05(2023)058

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Abstract We present a framework to study the entanglement structure of a quantum field theory inspired by the formalism of particle detectors in relativistic quantum information. This framework can in principle be used to faithfully capture entanglement in a QFT between arbitrary-shaped regions of spacetime without encountering UV divergences, bypassing many of the issues typically present in other approaches. Our results also establish the limits of the efficiency of entanglement harvesting, and may also be used to motivate an operational definition of entanglement between spacetime subregions in field theory.
We study the protocol of entanglement harvesting when the particle detectors that harvest entanglement from the field are replaced by fully relativistic quantum field theories. We show that two localized modes of the quantum field theories are able to harvest the same amount of leading order entanglement as two nonrelativistic particle detectors, thus implying that quantum field theoretical probes can generally harvest more entanglement than particle detectors. These results legitimize the use of particle detectors to study entanglement harvesting regardless of their internally nonrelativistic nature.
Entanglement harvesting provides an operational means of probing the nonlocal correlation structure of quantum fields through localized detector-field interactions. Although harvesting has been investigated in a wide variety of spacetime backgrounds and quantum field states, existing studies largely focus on individual detector configurations or specific physical settings. In this work, we develop a unified perturbative framework for entanglement harvesting with localized Unruh-DeWitt detectors that is applicable to arbitrary detector trajectories and quantum field states through the corresponding Wightman function. By expressing the detector response in terms of dimensionless variables and analytically reducing the perturbative integrals to numerically efficient representations, the framework enables a systematic exploration of multidimensional harvesting landscapes, phase boundaries, and harvesting robustness. As representative applications, we investigate inertial detectors in the Minkowski vacuum, thermal field states, and uniformly accelerated observers. We show that finite temperature enhances both the local detector response and the nonlocal exchange amplitude, leading, within the perturbative Gaussian-switching framework considered here, to an enlargement of the harvesting region in parameter space. The resulting harvesting landscapes provide a global characterization of detector-field interactions and demonstrate how temperature and observer motion reshape the competition between local detector noise and nonlocal field correlations. The framework developed here provides a unified and readily extensible methodology for investigating entanglement harvesting across a broad class of relativistic quantum systems.
Entanglement has long been the subject of discussion by philosophers of quantum theory, and has recently come to play an essential role for physicists in their development of quantum information theory. In this paper we show how the formalism of algebraic quantum field theory (AQFT) provides a rigorous framework within which to analyze entanglement in the context of a fully relativistic formulation of quantum theory. What emerges from the analysis are new practical and theoretical limitations on an experimenter’s ability to perform operations on a field in one spacetime region that can disentangle its state from the state of the field in other spacelike-separated regions. These limitations show just how deeply entrenched entanglement is in relativistic quantum field theory, and yield a fresh perspective on the ways in which the theory differs conceptually from both standard nonrelativistic quantum theory and classical relativistic field theory. “...despite its conservative way of dealing with physical principles, algebraic QFT leads to a radical change of paradigm. Instead of the Newtonian view of a space-time filled with a material content one enters the reality of Leibniz created by relation (in particular inclusions) between ‘monads ’ (∼ the hyperfinite type III1 local von Neumann factors A(O) which as single algebras are nearly void of physical meaning) ” Schroer (1998, p. 302).
We present a comparison of the AQFT-based Fewster-Verch framework with the Unruh-DeWitt particle detector models commonly employed in relativistic quantum information and QFT in curved space. We use this comparison to respond to a recent paper [M. H. Ruep, Weakly coupled local particle detectors cannot harvest entanglement, arXiv:2103.13400.] in which it was argued that the Reeh-Schlieder theorem prevents weakly coupled local particle detectors from harvesting vacuum entanglement from a quantum field. Their claim can be traced back to the mixedness that a local particle detector cannot escape from because of the entanglement that it must have with the quantum fields outside of the localization area. We argue that for any realistic scale of localization for physical particle detectors, the effect of that mixedness is negligible, and that weakly coupled, localized particle detectors are not impeded from harvesting vacuum entanglement.
Abstract Many states of linear real scalar quantum fields (in particular Reeh–Schlieder states) on flat as well as curved spacetime are entangled on spacelike separated local algebras of observables. It has been argued that this entanglement can be ‘harvested’ by a pair of so-called particle detectors, for example singularly or non-locally coupled quantum mechanical harmonic oscillator Unruh detectors. In an attempt to avoid such imperfect coupling, we analyse a model-independent local and covariant entanglement harvesting protocol based on the local probes of a recently proposed measurement theory of quantum fields. We then introduce the notion of a local particle detector concretely given by a local mode of a linear real scalar probe field on possibly curved spacetime and possibly under the influence of external fields. In a non-perturbative analysis we find that local particle detectors cannot harvest entanglement below a critical coupling strength when the corresponding probe fields are initially prepared in quasi-free Reeh–Schlieder states and are coupled to a system field prepared in a quasi-free state. This is a consequence of the fact that Reeh–Schlieder states restrict to truly mixed states on any local mode.
We present a fully relativistic model for localized probes in quantum field theory. Furthermore, we show that it is possible to obtain particle detector models from localized quantum field theories that interact with a free quantum field. In particular, a particle detector model is obtained when one traces out over inaccessible degrees of freedom of the localized field. This gives rise to a particle detector model, that is, a quantum degree of freedom that couples to a free field theory in an extended region of spacetime. Moreover, we show that the predictions of traditional particle detector models and fully relativistic localized fields completely coincide to leading order in perturbation theory.