Quantum-information notions such as the Local Operations and Classical Communication (LOCC) theorem presuppose a spacetime-local tensor-product factorization of the total Hilbert space, which gauge constraints destroy. Spalvieri, Garmier and Giacomini (arXiv:2512.19806, 2025; v2 2026) recently constructed, for a two-dimensional lattice gauge toy model of QED-like structure with point-like semiclassical sources, gauge-invariant local operator algebras and a sectorwise decomposition of the physical Hilbert space into superselection sectors labelled by edge data, within which an ordinary tensor product and a generalized LOCC theorem hold; they applied this to field-mediated entanglement (FME) protocols proposed as witnesses of the quantum nature of gravity. The open problem is the extension of this operational framework beyond the toy model, exactly as posed in the paper's Outlook: (i) to genuinely delocalized quantum sources with non-negligible position and momentum uncertainties; (ii) to the strict continuum, where the underlying operator algebras change from type I to type III and the discrete sector decomposition loses its direct meaning, requiring a precise characterization of when effective type-I operational structures emerge at finite experimental resolution; (iii) to a physical and operational interpretation of the edge data labelling the sectors, e.g. as edge modes or boundary reference frames; and (iv) to linearized quantum gravity, highlighted by the authors as the central open goal.

Background

In quantum information theory, the resource theory of entanglement is built on subsystem locality: the total Hilbert space is a tensor product HA⊗HB\mathcal{H}_A \otimes \mathcal{H}_B, each party acts by local operations on its factor, and they may share classical communication (together: LOCC). The LOCC theorem states that such LOCC protocols cannot create entanglement between AA and BB from a separable state. Gauge theories violate the premise: physical states must satisfy first-class constraints (Gauss laws) that tie field and source degrees of freedom across space, so the gauge-invariant physical Hilbert space does not factorize into a spacetime-local tensor product. A large literature (e.g. Van Acoleyen et al. 2016 on entanglement distillation in lattice gauge theories; Donnelly–Wall 2015, 2016 and Donnelly–Freidel 2016 on edge modes) analyzed entanglement in this setting, identifying superselection sectors — orthogonal blocks of the physical Hilbert space that local gauge-invariant operations cannot mix — and edge modes, i.e. boundary degrees of freedom arising from gauge transformations supported at the cut between regions. Dressing is the standard tool for making operators gauge-invariant at the cost of nonlocal support (e.g. attaching a Wilson line, the path-ordered exponential of the gauge field along a curve ending on the charge, to a charged field). Most such treatments, however, concern finite-dimensional models and do not give an operational account of local operations or entanglement generation in continuum gauge theories.

These questions became urgent through proposals to test the quantum nature of gravity via field-mediated entanglement (FME, also called gravitationally induced entanglement, GIE): two mesoscopic masses prepared in spatial superposition, allowed to interact only gravitationally, should become entangled (Bose et al. 2017; Marletto–Vedral 2017), and the LOCC theorem is invoked to conclude that the mediator must be nonclassical. Since gravity is a gauge theory, this reasoning is not straightforward (see also Christodoulou et al. 2023 on locally mediated entanglement in linearized quantum gravity). Spalvieri, Garmier and Giacomini (2025) addressed the gap with an explicit two-dimensional lattice gauge toy model capturing key structural features of QED. Diagonalizing the center of the gauge-invariant local algebra of a region AA, they obtained a decomposition of the physical Hilbert space into superselection sectors labelled by edge data (denoted KK), such that within each sector the Hilbert space factorizes as HAK⊗HAˉK\mathcal{H}_A^K \otimes \mathcal{H}_{\bar A}^K, and the LOCC theorem applies sectorwise — a generalized LOCC theorem for Hilbert spaces without a global local tensor product. They gave dressing prescriptions that create superpositions of sources in the presence of the constraints, applied the framework to an FME protocol, and ruled out an embezzlement (approximate-entanglement-generation) route. Their continuum limit (Appendix K) reproduces two-dimensional QED, but the von Neumann algebras of local regions (algebras of bounded operators on Hilbert space closed under weak limits) change type: the type-I algebras (which admit Hilbert-space tensor factorizations and ordinary density matrices) supporting the discrete sector decomposition become type-III in the continuum — the generic situation in algebraic quantum field theory (Yngvason 2005; Sorce 2024) — and type-III algebras admit no Hilbert-space tensor factorization at all.

The paper's Outlook (unchanged in the July 2026 v2 revision) lists four directions as open: extending to genuinely delocalized quantum sources (sources with non-negligible position and momentum uncertainties, relevant for gravitational fields sourced by delocalized quantum states as studied by Chen–Giacomini–Rovelli 2023 and Chen–Giacomini 2025, which will require dressing prescriptions accounting for the source's quantum state and back-reaction); making precise when and under which limiting procedures effective type-I operational structures emerge from the fundamentally type-III continuum theory, e.g. by reducing the lattice spacing only to a finite, resolution-dependent scale; clarifying the physical interpretation and operational role of the edge terms labelling the sectors — their suggested relation to edge modes (Donnelly–Wall, Donnelly–Freidel) and to quantum reference frames (Carrozza–Höhn 2022, who recast edge modes as reference frames obtained by post-selection; Kabel–Brukner–Wieland 2023 on quantum reference frames at spacetime boundaries; a quantum reference frame is a physical system whose degrees of freedom are used to define relations to other systems, replacing external classical frames); and, as the central open goal, extending the whole construction — gauge-invariant local algebras, sectorwise Hilbert-space decomposition, dressing prescriptions, generalized LOCC theorem — to linearized quantum gravity, where additional obstructions from the gravitational gauge symmetry (diffeomorphisms rather than an internal symmetry) may arise (cf. Boulle–Franzmann 2026; Aziz–Howl 2025).

Problem Statement

Extend the operational framework of Spalvieri–Garmier–Giacomini — gauge-invariant local operator algebras, the sectorwise decomposition of the physical Hilbert space into superselection sectors labelled by edge data (with a tensor-product structure within each sector), dressing prescriptions for creating superpositions of sources, and the resulting generalized LOCC theorem — beyond their two-dimensional lattice gauge toy model, in the four linked directions the paper itself poses as its program toward 'a comprehensive operational formulation of quantum gauge fields in an information-theoretic language':

  1. Delocalized quantum sources. Replace the semiclassical, approximately localized qubit matter by sources with non-negligible position and momentum uncertainties, requiring dressing prescriptions and an operational description of local operations that consistently account for the source's quantum state and its back-reaction on the field (the regime relevant to gravitational fields sourced by delocalized quantum states).

  2. Continuum limit and the type-I/type-III transition. The discrete sectorwise decomposition is supported by type-I von Neumann algebras that become type-III in the strict continuum, where no Hilbert-space tensor factorization exists. Characterize precisely when and under which limiting procedures — e.g. reducing the lattice spacing only to a finite, resolution-dependent scale — effective type-I operational structures emerge from the fundamentally type-III continuum theory, such that the sectorwise local tensor product and the operational LOCC framework remain valid; make this finite-resolution argument precise.

  3. Operational meaning of the edge data. Clarify the physical interpretation and operational role of the edge terms labelling the superselection sectors: their relation to edge modes in gauge theories with boundaries and to quantum-reference-frame structures, and the physical process by which measuring or fixing the edge data projects onto (or selects) a superselection sector, together with the consequences for applying the generalized LOCC theorem in concrete experimental scenarios.

  4. Linearized quantum gravity (central goal). Construct explicitly, for linearized quantum gravity coupled to quantum matter, the gauge-invariant local algebras, the sectorwise Hilbert-space decomposition with its edge data, the dressing prescriptions creating superpositions of gravitating sources, and the corresponding generalized LOCC theorem; identify any additional obstructions unique to gravitational gauge symmetry (diffeomorphism constraints rather than internal gauge symmetry).

These components are jointly needed milestones of one objective: a fully operational account of locality and entanglement generation in continuum gauge theory culminating in the gravitational case. The gravity construction is the completion boundary; components 1–3 enter as constitutive prerequisites or clarifications that the gravitational construction must handle or sharply delimit (delocalized matter is required for realistic FME sources; the finite-resolution type-I analysis is required to interpret the construction as a genuine continuum-field statement; the operational role of the edge data is required to apply the sectorwise LOCC theorem in experimentally meaningful terms). Progress on individual components, or a rigorous demonstration that a component is obstructed, constitutes partial progress on the problem as posed.

References

  1. A. Spalvieri, S. C. Garmier, F. Giacomini (2025). Local Operations and Field Mediated Entanglement without a Local Tensor Product Structure. arXiv:2512.19806, DOI: 10.48550/arXiv.2512.19806. https://arxiv.org/abs/2512.19806
  2. S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim, G. Milburn (2017). Spin entanglement witness for quantum gravity. Physical Review Letters 119, 240401. DOI: 10.1103/PhysRevLett.119.240401. https://doi.org/10.1103/PhysRevLett.119.240401
  3. C. Marletto, V. Vedral (2017). Gravitationally induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity. Physical Review Letters 119, 240402. DOI: 10.1103/PhysRevLett.119.240402. https://doi.org/10.1103/PhysRevLett.119.240402
  4. M. Christodoulou, A. Di Biagio, M. Aspelmeyer, Č. Brukner, C. Rovelli, R. Howl (2023). Locally mediated entanglement in linearized quantum gravity. Physical Review Letters 130, 100202. DOI: 10.1103/PhysRevLett.130.100202. https://doi.org/10.1103/PhysRevLett.130.100202
  5. L. Chen, F. Giacomini, C. Rovelli (2023). Quantum States of Fields for Quantum Split Sources. Quantum 7, 958. DOI: 10.22331/q-2023-03-20-958. https://doi.org/10.22331/q-2023-03-20-958
  6. L. Chen, F. Giacomini (2025). Quantum Effects in Gravity Beyond the Newton Potential from a Delocalized Quantum Source. Physical Review X 15, 031063. DOI: 10.1103/hl1c-t8z9. https://doi.org/10.1103/hl1c-t8z9
  7. S. Carrozza, P. A. Höhn (2022). Edge modes as reference frames and boundary actions from post-selection. Journal of High Energy Physics 2022, 172. DOI: 10.1007/JHEP02(2022)172. https://doi.org/10.1007/JHEP02(2022)172
  8. V. Kabel, Č. Brukner, W. Wieland (2023). Quantum reference frames at the boundary of spacetime. Physical Review D 108, 106022. DOI: 10.1103/PhysRevD.108.106022. https://doi.org/10.1103/PhysRevD.108.106022
  9. W. Donnelly, L. Freidel (2016). Local subsystems in gauge theory and gravity. Journal of High Energy Physics 2016, 102. arXiv:1601.04744, DOI: 10.1007/JHEP09(2016)102. https://doi.org/10.1007/JHEP09(2016)102
  10. W. Donnelly, A. C. Wall (2015). Entanglement entropy of electromagnetic edge modes. Physical Review Letters 114, 111603. DOI: 10.1103/PhysRevLett.114.111603. https://doi.org/10.1103/PhysRevLett.114.111603
  11. W. Donnelly, A. C. Wall (2016). Geometric entropy and edge modes of the electromagnetic field. Physical Review D 94, 104053. DOI: 10.1103/PhysRevD.94.104053. https://doi.org/10.1103/PhysRevD.94.104053
  12. K. Van Acoleyen, N. Bultinck, J. Haegeman, M. Marien, V. B. Scholz, F. Verstraete (2016). Entanglement of Distillation for Lattice Gauge Theories. Physical Review Letters 117, 131602. DOI: 10.1103/PhysRevLett.117.131602. https://doi.org/10.1103/PhysRevLett.117.131602
  13. J. Yngvason (2005). The role of type III factors in quantum field theory. Reports on Mathematical Physics 55, 135–147. DOI: 10.1016/S0034-4877(05)80009-6. https://doi.org/10.1016/S0034-4877(05)80009-6
  14. J. Sorce (2024). Notes on the type classification of von Neumann algebras. Reviews in Mathematical Physics 36, 2430002. arXiv:2302.01958, DOI: 10.1142/S0129055X24300024. https://doi.org/10.1142/S0129055X24300024
  15. N. Boulle, G. Franzmann (2026). Subsystems independence in gravitationally induced entanglement proposals. Physical Review D 114, 025015. arXiv:2512.17024, DOI: 10.1103/tgjx-y7hl. https://doi.org/10.1103/tgjx-y7hl
  16. S. L. Ludescher, L. D. Loveridge, T. D. Galley, M. P. Müller (2026). Gravity-mediated entanglement via infinite-dimensional systems. Journal of Physics A: Mathematical and Theoretical 59, 215302. arXiv:2507.13201, DOI: 10.1088/1751-8121/ae65e7. https://doi.org/10.1088/1751-8121/ae65e7
  17. J. Yant, M. Blencowe (2026). Operational quantum field theoretic model for gravitationally induced entanglement. Physical Review D 114, 026006. arXiv:2503.20855, DOI: 10.1103/rvky-5ppb. https://doi.org/10.1103/rvky-5ppb
  18. J. Aziz, R. Howl (2025). Classical theories of gravity produce entanglement. Nature 646, 813–817. DOI: 10.1038/s41586-025-09595-7. https://doi.org/10.1038/s41586-025-09595-7
  19. A. Di Biagio (2025). The simple reason why classical gravity can entangle. arXiv:2511.02683. https://arxiv.org/abs/2511.02683
  20. E. Bianchi, P. Donà, E. Muiño (2026). Typical entanglement entropy with charge conservation. arXiv:2604.26141. https://arxiv.org/abs/2604.26141