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arXiv:2411.13605 · Letters in Mathematical Physics

Quantum field measurements in the Fewster–Verch framework

Jan Mandrysch, Miguel Navascués

Published Oct 24, 2025• 5 citations
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Abstract

Abstract The Fewster–Verch (FV) framework provides a local and covariant approach for defining measurements in quantum field theory (QFT). Within this framework, a probe QFT represents the measurement device, which, after interacting with the target QFT, undergoes an arbitrary local measurement. Remarkably, the FV framework is free from Sorkin-like causal paradoxes and robust enough to enable quantum state tomography. However, two open issues remain. First, it is unclear if the FV framework allows conducting arbitrary local measurements. Second, if the probe field is interpreted as physical and the FV framework as fundamental, then one must demand the probe measurement to be itself implementable within the framework. That would involve a new probe, which should also be subject to an FV measurement, and so on. It is unknown if there exist non-trivial FV measurements for which such an “FV-Heisenberg cut” can be moved arbitrarily far away. In this work, we advance the first problem by proving that Gaussian-modulated measurements of locally smeared fields fit within the FV framework. We solve the second problem by showing that any such measurement admits a movable FV-Heisenberg cut. As a technical by-product, we establish that state transformations induced by finite-rank perturbations of the classical phase space underlying a linear scalar field preserve the Hadamard property.

DOI
10.1007/s11005-025-02001-3

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Abstract The Fewster–Verch (FV) framework provides a local and covariant approach for defining measurements in quantum field theory (QFT). Within this framework, a probe QFT represents the measurement device, which, after interacting with the target QFT, undergoes an arbitrary local measurement. Remarkably, the FV framework is free from Sorkin-like causal paradoxes and robust enough to enable quantum state tomography. However, two open issues remain. First, it is unclear if the FV framework allows conducting arbitrary local measurements. Second, if the probe field is interpreted as physical and the FV framework as fundamental, then one must demand the probe measurement to be itself implementable within the framework. That would involve a new probe, which should also be subject to an FV measurement, and so on. It is unknown if there exist non-trivial FV measurements for which such an “FV-Heisenberg cut” can be moved arbitrarily far away. In this work, we advance the first problem by proving that Gaussian-modulated measurements of locally smeared fields fit within the FV framework. We solve the second problem by showing that any such measurement admits a movable FV-Heisenberg cut. As a technical by-product, we establish that state transformations induced by finite-rank perturbations of the classical phase space underlying a linear scalar field preserve the Hadamard property.
As noted by Sorkin, regarding quantum instruments whose Kraus operators are localizable within some spacetime region as operations accessible therein leads to superluminal communication. This so-called Sorkin paradox can be resolved by further constraining the set of allowed local operations in quantum field theory (QFT). In this spirit, Fewster and Verch proposed a framework for local QFT operations that generalizes non-relativistic quantum measurement theory and does not lead to Sorkin-like paradoxes. Shortly afterwards, Jubb (and later Oeckl) identified the minimal conditions that QFT instruments must satisfy to be compatible with Einstein's causality. In this work, we study both approaches in the quantum field theory of the free scalar field. First, we prove that a very wide class of causal instruments is FV-realizable: namely, those whose measurement channels are random displacements of the field operators. As we show, this class allows implementing arbitrary instruments in a heralded, probabilistic way, as well as non-demolition measurements deterministically. Second, we construct examples of causal channels that do not admit an approximate FV realization. Some of such causal, not FV-realizable channels violate basic physical principles, so they should not be part of any measurement theory for QFT. Third, we investigate the difficulty of characterizing the set of QFT channels that can be generated through the composition of several FV schemes. In this regard, we find a countable family of simple QFT channels for which no Turing machine can discriminate between channels within or far away from the implementable set.
Abstract Arguments by Sorkin (Impossible measurements on quantum fields. In: Directions in general relativity: proceedings of the 1993 International Symposium, Maryland, vol 2, pp 293–305, 1993) and Borsten et al. (Phys Rev D 104(2), 2021. https://doi.org/10.1103/PhysRevD.104.025012 ) establish that a natural extension of quantum measurement theory from non-relativistic quantum mechanics to relativistic quantum theory leads to the unacceptable consequence that expectation values in one region depend on which unitary operation is performed in a spacelike separated region. Sorkin [1] labels such scenarios ‘impossible measurements’. We explicitly present these arguments as a no-go result with the logical form of a reductio argument and investigate the consequences for measurement in quantum field theory (QFT). Sorkin-type impossible measurement scenarios clearly illustrate the moral that Microcausality is not by itself sufficient to rule out superluminal signalling in relativistic quantum theories that use Lüders’ rule. We review three different approaches to formulating an account of measurement for QFT and analyze their responses to the ‘impossible measurements’ problem. Two of the approaches are: a measurement theory based on detector models proposed in Polo-Gómez et al. (Phys Rev D, 2022. https://doi.org/10.1103/physrevd.105.065003 ) and a measurement framework for algebraic QFT proposed in Fewster and Verch (Commun Math Phys 378(2):851–889, 2020). Of particular interest for foundations of QFT is that they share common features that may hold general morals about how to represent measurement in QFT. These morals are about the role that dynamics plays in eliminating ‘impossible measurements’, the abandonment of the operational interpretation of local algebras A(O){\mathcal {A}}(O)A(O) A ( O ) as representing possible operations carried out in region O , and the interpretation of state update rules. Finally, we examine the form that the ‘impossible measurements’ problem takes in histories-based approaches and we discuss the remaining challenges.
This paper is part of a broader project on the Hartmann Operator Geometry (HTSD) developed in the monograph “The Hartmann Operator Geometry – Volume I”. This preprint investigates the reduced Heisenberg evolution induced by local system–probe measurements in the Fewster–Verch framework of locally covariant quantum field theory. Starting from locally covariant system and probe functors and a local coupling in a compact region, the work considers the induced completely positive unital maps on the system algebra and assumes a weak–coupling / short–time regime that yields a norm–continuous Markovian semigroup. Under this local Markovianity assumption, the generator of the reduced dynamics is shown to have a purely diffusion–type Lindblad form built from finitely many selfadjoint “pointer” observables. Each pointer observable defines a local inner derivation, and the generator can be written as a Laplace–type operator of HTSD form, i.e. as a quadratic combination of these derivations. In this way, HTSD operator geometry appears as a natural differential structure underlying local measurement dynamics in locally covariant AQFT, with an associated Dirichlet form and curvature tensor encoding energy and non-integrability properties of the measurement–induced evolution.
Abstract The process of quantum measurement is considered in the algebraic framework of quantum field theory on curved spacetimes. Measurements are carried out on one quantum field theory, the “system”, using another, the “probe”. The measurement process involves a dynamical coupling of “system” and “probe” within a bounded spacetime region. The resulting “coupled theory” determines a scattering map on the uncoupled combination of the “system” and “probe” by reference to natural “in” and “out” spacetime regions. No specific interaction is assumed and all constructions are local and covariant. Given any initial state of the probe in the “in” region, the scattering map determines a completely positive map from “probe” observables in the “out” region to “induced system observables”, thus providing a measurement scheme for the latter. It is shown that the induced system observables may be localized in the causal hull of the interaction coupling region and are typically less sharp than the probe observable, but more sharp than the actual measurement on the coupled theory. Post-selected states conditioned on measurement outcomes are obtained using Davies–Lewis instruments that depend on the initial probe state. Composite measurements involving causally ordered coupling regions are also considered. Provided that the scattering map obeys a causal factorization property, the causally ordered composition of the individual instruments coincides with the composite instrument; in particular, the instruments may be combined in either order if the coupling regions are causally disjoint. This is the central consistency property of the proposed framework. The general concepts and results are illustrated by an example in which both “system” and “probe” are quantized linear scalar fields, coupled by a quadratic interaction term with compact spacetime support. System observables induced by simple probe observables are calculated exactly, for sufficiently weak coupling, and compared with first order perturbation theory.
The Summers-Werner theorem shows that the vacuum of a relativistic quantum field theory violates the Bell-CHSH inequality maximally on complementary wedges, but it is an existence statement: it exhibits neither the observables nor an apparatus that measures them. We give an explicit operator-level realization within the Fewster-Verch measurement framework, obtaining dichotomic Bell observables of a free Majorana field in 1+11+11+1 dimensions as observables induced by local couplings to Majorana probes. The induced system smearing factorizes exactly as fh=k bλ,hf_h=k\,b_{λ,h}fh​=kbλ,h​, so its support is controlled by the compactly supported coupling profile kkk; dichotomy is imposed by calibrating and normalizing the induced smearings rather than required of the probe observable; and a probe-only normal ordering yields the product of Alice's and Bob's induced bilinears, which the non-multiplicative induced map alone does not provide. Under explicit attainability hypotheses the induced observables saturate the Tsirelson bound in a state prepared from the vacuum. The construction extends to coupled scalar fields.
Quantum operations that are perfectly admissible in non-relativistic quantum theory can enable signalling between spacelike separated regions when naively imported into quantum field theory (QFT). Prominent examples of such "impossible measurements" include certain unitary kicks and projective measurements. It is generally accepted that only those quantum operations whose physical implementation arises from a fully relativistically covariant interaction, between the quantum field and a suitable probe, should be regarded as admissible. While this idea has been realised at the level of abstract algebraic QFT, or via particular measurement models, there is still no general set of operational criteria characterising which measurements are physically implementable. In this work we adopt the local S-matrix formalism to describe local operations, and make use of a hierarchy of factorisation conditions that exclude both superluminal signalling and retrocausality, thereby providing such a criterion. Realising the local S-matrices through explicit interactions between smeared field operators and a pointer degree of freedom, we further derive local causality conditions for the induced Kraus operators, which guarantee the absence of signalling in "impossible measurement" scenarios. Finally, we show that the accuracy with which local field observables can be measured is fundamentally limited by the retarded propagator of the field, which plays an essential role in the factorisation identity we prove for the field Kraus operators.