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Non-Factorable Phase Structure as a Necessary and Sufficient Condition for Gravity-Mediated Entanglement

Cary Miller

Published Feb 4, 2026• 0 citations
Publisher

Abstract

This work presents a model-independent analysis of gravity-mediated entanglement based on the geometric structure of interaction-induced phase evolution. Using a minimal two-body phase model inspired by recent gravity-mediated entanglement proposals, it is shown that quantum entanglement between initially separable systems arises if and only if the induced phase structure is non-factorable. The central result identifies a gauge-invariant plaquette phase as a necessary and sufficient operational condition for entanglement generation. When this plaquette invariant is nonzero, standard quantum information diagnostics—such as negativity, concurrence, and Bell–CHSH correlations—confirm the presence of entanglement. When the invariant is eliminated, entanglement vanishes identically, even when individual phase magnitudes remain nonzero. A comprehensive suite of falsification tests is applied, including local phase factorization, plaquette removal, deterministic and stochastic phase scrambling, permutation symmetry tests, LOCC protocols, causal reordering, and controlled noise injection. Across all tests, entanglement survives exclusively in the presence of non-factorable phase structure. Scaling analyses further demonstrate consistent dependencies on mass, interaction time, separation distance, and superposition size, yielding a distinctive interaction fingerprint incompatible with classical or semiclassical channels. The framework is fully operational and does not assume a specific theory of quantum gravity. Instead, it provides a concrete, experimentally accessible criterion for probing nonclassical aspects of gravity using near-term interferometric platforms. The results sharpen the interpretational foundations of gravity-mediated entanglement proposals by isolating the minimal structural feature responsible for entanglement generation.

DOI
10.5281/zenodo.18478847

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Curvature-Induced Phase Loops as the Geometric Origin of Gravity-Mediated Entanglement

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This paper establishes a geometric bridge between two operational results concerning gravity-mediated entanglement and phase-based field structure. First, prior work showed that entanglement in minimal gravity-mediated interferometric models arises if and only if the interaction phases contain a non-factorable plaquette invariant. This invariant represents an oriented loop phase that cannot be decomposed into local contributions and therefore serves as a necessary and sufficient condition for entanglement generation. Second, the Harmonic Code framework describes persistent physical effects as loop-structured phase geometry, in which closed phase loops encode enduring information about curvature and field structure. The present work unifies these viewpoints by demonstrating that the plaquette phase emerges naturally as a curvature-induced loop holonomy. An effective phase-gradient field is introduced, and the oriented loop phase is defined as Π = ∮ k · dℓ = ∬ (∇ × k) · dA, showing that the non-factorable phase invariant is directly generated by the curl of the phase-gradient field. A minimal rotational field model is used to derive the relation Π = 2εA, where A is the enclosed area and ε is the curl parameter. Numerical simulations verify the key geometric properties expected of a true holonomy: Linear scaling with enclosed area Shape independence at fixed area Sign reversal under orientation flip Convergence under discretization refinement The loop phase Π is then mapped to standard two-qubit entanglement measures. For the separable input state |++⟩, the concurrence and negativity are shown to depend only on Π: C(Π) = |sin(Π/2)| N(Π) = C/2 These results demonstrate that the non-factorable phase structure required for gravity-mediated entanglement has a direct geometric origin as a curvature-induced loop holonomy. The paper therefore provides a unified operational statement: Curvature generates loop phases, and those loop phases are exactly the structures responsible for gravity-mediated entanglement. This establishes a concrete geometric mechanism linking spacetime curvature, phase holonomy, and quantum entanglement, and offers a clear, experimentally testable framework for probing quantum aspects of gravity.

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Newtonian Gravitational Phase, Plaquette Invariants, and Entanglement: An Operational Test of Gravity-Mediated Quantum Correlations

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GRAVITY AS A QUANTUM MEDIATOR AT THE MESOSCOPIC INTERFACE: Channel Structure, Entanglement Generation, and the Structural Failure of Semiclassical Gravity in BMV-Type Experiments

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ABSTRACT Background The gravity–quantum interface lacks a minimal, operational theory for the weak-field, non-relativistic, mesoscopic regime. Semiclassical gravity (SCG), which sources the classical Einstein equations from the quantum expectation value of stress–energy, is the dominant approximation but was designed for cosmological and astrophysical contexts, not for two-body entanglement experiments in the laboratory. A structural silence exists: no framework specifies how gravity couples to quantum matter when entanglement between spatially delocalised mesoscopic masses is at stake. Gap The Named Binary distinguishing Semiclassical Gravity Interface (SCGI) from Quantum-Mediator Gravity Interface (QMGI) does not appear in the literature as a formal operational distinction with computable consequences. No framework simultaneously satisfies: standard quantum mechanics for matter, no-signalling, reduction to Newtonian gravity for classical configurations, and entanglement-generation capability. This four-constraint conjunction defines the structural gap. Approach We formalise the two-body BMV (Bose–Marletto–Vedral) configuration using Hilbert-space language and quantum channels, derive the effective entangling Hamiltonian, prove that C(t) = |sin(Δφ(t)/2)| where Δφ(t) is the entangling phase determined by geometry and interaction time, and derive the design inequality τ_ent ≪ τ_dec as the admissibility condition for entanglement detection. We establish the Strongest Formulation in the five-part programme template, specify a pre-registerable CCS with quantitative decision rule, confirm structural invariance across three independent physics domains, ground the framework in IGT theory, and provide a complete Weil Protocol practitioner review pack. Results Under the four-constraint conjunction (standard QM + no-signalling + Newtonian reduction + entanglement capability), gravity in the mesoscopic weak-field regime must be modelled as a quantum channel: SCGI fails because classical channels cannot increase entanglement between initially separable systems. The entangling phase scales as Δφ(t) ~ Gm²dt/(ℏD²); detectable entanglement requires τ_ent/τ_dec < 1. Three independent distinguishing predictions follow that SCGI cannot make. Structural invariance is confirmed in quantum electrodynamics (photon-mediated entanglement), nuclear physics (exchange-mediated forces), and quantum optics (cavity-mediated entanglement). Implications Any tabletop detection of gravitationally induced entanglement between two mesoscopic masses in a BMV-type experiment would constitute a direct, model-independent, low-energy signature of the quantum nature of gravity. Any confirmed null result provides the strongest direct evidence for SCGI-type models and bounds the parameter space of quantum mediator theories. Weil Protocol practitioner review is required before policy adoption of experimental specifications; status INCOMPLETE. Human-stakes level: L2 · Weil Protocol: INCOMPLETE

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Relativistic Gravity-Induced Entanglement via Frame Dragging

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Classical timing noise in gravity-mediated entanglement tests: LOCC structure, metrological bounds, and visibility thresholds

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Table-top proposals to test gravity-mediated entanglement aim to distinguish coherent gravitational interactions from classical dephasing processes that generate identical phases on both interferometers. A particularly important contribution is platform-invariant timing noise, which can be accessed through optical-clock cross-spectra and frequency-transfer links. In this work we (i) derive the mapping from clock cross-spectra to an effective common-mode dephasing rate, (ii) show that the corresponding dynamical channel is LOCC and therefore unable to generate entanglement, and (iii) combine published clock and interferometer noise floors to obtain quantitative bounds on the ratio between common-mode and local dephasing rates in representative gravity-entanglement proposals. Across QGEM-, MAQRO-, and levitated-nanoparticle regimes we find a robust hierarchy $γ_{\rm com}/γ_{\rm loc}\sim10^{-9}$--$10^{-11}$, identifying platform-invariant timing noise as a negligible but calibratable background. We further derive visibility and Bell--CHSH thresholds in the presence of both common-mode and local dephasing, and illustrate the full calibration workflow with a simple synthetic example anchored to state-of-the-art clock metrology. Extended derivations and statistical tools are provided in the Supplemental Material.

arXiv:2511.11879•Nov 14, 2025•0 citations
This work presents a model-independent analysis of gravity-mediated entanglement based on the geometric structure of interaction-induced phase evolution. Using a minimal two-body phase model inspired by recent gravity-mediated entanglement proposals, it is shown that quantum entanglement between initially separable systems arises if and only if the induced phase structure is non-factorable. The central result identifies a gauge-invariant plaquette phase as a necessary and sufficient operational condition for entanglement generation. When this plaquette invariant is nonzero, standard quantum information diagnostics—such as negativity, concurrence, and Bell–CHSH correlations—confirm the presence of entanglement. When the invariant is eliminated, entanglement vanishes identically, even when individual phase magnitudes remain nonzero. A comprehensive suite of falsification tests is applied, including local phase factorization, plaquette removal, deterministic and stochastic phase scrambling, permutation symmetry tests, LOCC protocols, causal reordering, and controlled noise injection. Across all tests, entanglement survives exclusively in the presence of non-factorable phase structure. Scaling analyses further demonstrate consistent dependencies on mass, interaction time, separation distance, and superposition size, yielding a distinctive interaction fingerprint incompatible with classical or semiclassical channels. The framework is fully operational and does not assume a specific theory of quantum gravity. Instead, it provides a concrete, experimentally accessible criterion for probing nonclassical aspects of gravity using near-term interferometric platforms. The results sharpen the interpretational foundations of gravity-mediated entanglement proposals by isolating the minimal structural feature responsible for entanglement generation.
This paper establishes a geometric bridge between two operational results concerning gravity-mediated entanglement and phase-based field structure. First, prior work showed that entanglement in minimal gravity-mediated interferometric models arises if and only if the interaction phases contain a non-factorable plaquette invariant. This invariant represents an oriented loop phase that cannot be decomposed into local contributions and therefore serves as a necessary and sufficient condition for entanglement generation. Second, the Harmonic Code framework describes persistent physical effects as loop-structured phase geometry, in which closed phase loops encode enduring information about curvature and field structure. The present work unifies these viewpoints by demonstrating that the plaquette phase emerges naturally as a curvature-induced loop holonomy. An effective phase-gradient field is introduced, and the oriented loop phase is defined as Π = ∮ k · dℓ = ∬ (∇ × k) · dA, showing that the non-factorable phase invariant is directly generated by the curl of the phase-gradient field. A minimal rotational field model is used to derive the relation Π = 2εA, where A is the enclosed area and ε is the curl parameter. Numerical simulations verify the key geometric properties expected of a true holonomy: Linear scaling with enclosed area Shape independence at fixed area Sign reversal under orientation flip Convergence under discretization refinement The loop phase Π is then mapped to standard two-qubit entanglement measures. For the separable input state |++⟩, the concurrence and negativity are shown to depend only on Π: C(Π) = |sin(Π/2)| N(Π) = C/2 These results demonstrate that the non-factorable phase structure required for gravity-mediated entanglement has a direct geometric origin as a curvature-induced loop holonomy. The paper therefore provides a unified operational statement: Curvature generates loop phases, and those loop phases are exactly the structures responsible for gravity-mediated entanglement. This establishes a concrete geometric mechanism linking spacetime curvature, phase holonomy, and quantum entanglement, and offers a clear, experimentally testable framework for probing quantum aspects of gravity.
This work presents an explicit Newtonian gravitational phase model for two masses placed in spatial superposition and demonstrates that the resulting phase structure produces a non-zero plaquette invariant. Numerical simulations show that the plaquette phase scales with branch separation, generates entanglement according to analytic mappings, and vanishes under loop-removal kill tests. These results provide a fully operational demonstration that Newtonian gravitational phase accumulation produces the non-factorable loop structure required for gravity-mediated entanglement.
Quantum-gravity-induced entanglement of massive systems (QGEM) is commonly approximated in the nonrelativistic static limit by a Newtonian interaction between spatially separated masses. In this work, we reformulate the gravitationally mediated interaction phase in a conformally invariant twistor framework in which no notion of spacetime distance is assumed. We show that the bilocal phase responsible for entanglement generation remains well-defined and non-factorizable even in the absence of spacetime geometry. The familiar Newtonian 1/r1/r1/r phase, relevant for QGEM protocols, arises only after the conformal invariance is broken by introducing the infinity twistor, which selects a particular spacetime representation of the underlying bilocal quantum interaction. Our results isolate the genuinely quantum content of QGEM protocols and clarify the contingent role played by spacetime geometry in mediating entanglement.
ABSTRACT Background The gravity–quantum interface lacks a minimal, operational theory for the weak-field, non-relativistic, mesoscopic regime. Semiclassical gravity (SCG), which sources the classical Einstein equations from the quantum expectation value of stress–energy, is the dominant approximation but was designed for cosmological and astrophysical contexts, not for two-body entanglement experiments in the laboratory. A structural silence exists: no framework specifies how gravity couples to quantum matter when entanglement between spatially delocalised mesoscopic masses is at stake. Gap The Named Binary distinguishing Semiclassical Gravity Interface (SCGI) from Quantum-Mediator Gravity Interface (QMGI) does not appear in the literature as a formal operational distinction with computable consequences. No framework simultaneously satisfies: standard quantum mechanics for matter, no-signalling, reduction to Newtonian gravity for classical configurations, and entanglement-generation capability. This four-constraint conjunction defines the structural gap. Approach We formalise the two-body BMV (Bose–Marletto–Vedral) configuration using Hilbert-space language and quantum channels, derive the effective entangling Hamiltonian, prove that C(t) = |sin(Δφ(t)/2)| where Δφ(t) is the entangling phase determined by geometry and interaction time, and derive the design inequality τ_ent ≪ τ_dec as the admissibility condition for entanglement detection. We establish the Strongest Formulation in the five-part programme template, specify a pre-registerable CCS with quantitative decision rule, confirm structural invariance across three independent physics domains, ground the framework in IGT theory, and provide a complete Weil Protocol practitioner review pack. Results Under the four-constraint conjunction (standard QM + no-signalling + Newtonian reduction + entanglement capability), gravity in the mesoscopic weak-field regime must be modelled as a quantum channel: SCGI fails because classical channels cannot increase entanglement between initially separable systems. The entangling phase scales as Δφ(t) ~ Gm²dt/(ℏD²); detectable entanglement requires τ_ent/τ_dec < 1. Three independent distinguishing predictions follow that SCGI cannot make. Structural invariance is confirmed in quantum electrodynamics (photon-mediated entanglement), nuclear physics (exchange-mediated forces), and quantum optics (cavity-mediated entanglement). Implications Any tabletop detection of gravitationally induced entanglement between two mesoscopic masses in a BMV-type experiment would constitute a direct, model-independent, low-energy signature of the quantum nature of gravity. Any confirmed null result provides the strongest direct evidence for SCGI-type models and bounds the parameter space of quantum mediator theories. Weil Protocol practitioner review is required before policy adoption of experimental specifications; status INCOMPLETE. Human-stakes level: L2 · Weil Protocol: INCOMPLETE
Gravity-induced entanglement has been proposed as a method for testing the non-classical nature of gravity via tabletop experiments. While most existing proposals are restricted to the Newtonian limit, the frame dragging effect offers access to genuinely post-Newtonian features of the gravitational interaction and remains comparatively less explored. Here, we study gravity-induced entanglement generated by frame dragging in an interferometric setting and compute the entanglement phase between the rotational degrees of freedom of a source mass and the paths of a particle in two complementary ways: (i) via Schrödinger evolution with a quantized Lense-Thirring Hamiltonian in the large angular momentum limit, and (ii) via the on-shell action of linearized quantum gravity within the stationary phase approximation. Both approaches yield the same entanglement phase, consistent with the proper time difference between the interferometer arms. The path integral derivation further reveals how gravitational retardation modifies the entanglement phase, thereby making the local, relativistically causal linearized-gravity description explicit. Under the standard locality/mediator assumptions used in existing arguments, the resulting entanglement would witness non-classicality of the gravitational interaction.
Table-top proposals to test gravity-mediated entanglement aim to distinguish coherent gravitational interactions from classical dephasing processes that generate identical phases on both interferometers. A particularly important contribution is platform-invariant timing noise, which can be accessed through optical-clock cross-spectra and frequency-transfer links. In this work we (i) derive the mapping from clock cross-spectra to an effective common-mode dephasing rate, (ii) show that the corresponding dynamical channel is LOCC and therefore unable to generate entanglement, and (iii) combine published clock and interferometer noise floors to obtain quantitative bounds on the ratio between common-mode and local dephasing rates in representative gravity-entanglement proposals. Across QGEM-, MAQRO-, and levitated-nanoparticle regimes we find a robust hierarchy γcom/γloc∼10−9γ_{\rm com}/γ_{\rm loc}\sim10^{-9}γcom​/γloc​∼10−9--10−1110^{-11}10−11, identifying platform-invariant timing noise as a negligible but calibratable background. We further derive visibility and Bell--CHSH thresholds in the presence of both common-mode and local dephasing, and illustrate the full calibration workflow with a simple synthetic example anchored to state-of-the-art clock metrology. Extended derivations and statistical tools are provided in the Supplemental Material.