Tabletop Tests of Quantum Gravity Mediation

Two nearby masses can acquire a branch-dependent Newtonian phase while each occupies a spatial superposition. If that phase produces entanglement and every nongravitational channel is excluded, the experiment would probe whether gravity can transmit quantum information. The last inference is conditional: a positive witness is evidence for a nonclassical gravitational mediator only within a declared locality, composition, and mediation framework.

Required background. Equivalence Principles and Gravitational Measurement supplies the distinction between an invariant relative phase and a coordinate-dependent potential.

Helpful background. Flat-Spacetime Quantum Field Theory supplies matter-field mediation; Einstein Equations supplies the weak-field limit; Quantum Information supplies witnesses and channels; Quantum Gravity supplies the programme boundary; and Gravitational Radiation and Waveforms supplies calibrated inference habits.

Branch phases and the entangling invariant

Label the two localized branches of each mass by $i,j\in\{L,R\}$, and let their center-to-center separations during an interaction time $\tau$ be $d_{ij}$. In the point-mass, quasistatic limit, the four branch amplitudes acquire phases

$$\phi_{ij}=\frac{Gm_1m_2\tau}{d_{ij}}.$$

An overall phase and either mass’s local phase can be removed. The combination that cannot be removed locally is

$$\Phi=\phi_{LL}+\phi_{RR}-\phi_{LR}-\phi_{RL}.$$

Starting from equal branch superpositions, the ideal final state is

$$\lvert\psi\rangle =\frac12\sum_{i,j=L,R}e^{i\phi_{ij}}\lvert i\,j\rangle,$$

whose concurrence is

$$\mathcal C=\left\lvert\sin\frac{\Phi}{2}\right\rvert.$$

Thus gravity predicts entanglement when $\Phi$ is not an integer multiple of $2\pi$. For rigid finite-sized branches with mass densities normalized by $\int d^3x\,\rho_{a,i}(\mathbf x)=m_a$, replace the point-source factor by

$$\frac{m_1m_2}{d_{ij}} \longrightarrow \int d^3x\,d^3y\, \frac{\rho_{1,i}(\mathbf x)\rho_{2,j}(\mathbf y)} {\lvert\mathbf x-\mathbf y\rvert}.$$

A broad quantum wave packet also requires tracking variation of the phase across a branch; a single averaged phase need not describe its evolution. Imperfect recombination can suppress the accessible off-diagonal coherence. The original spin-tagged Stern–Gerlach proposal derives the branch phases and maps orbital coherence back to spin correlators in Bose et al. 2017, eqs. (1)–(7), pp. 2–4. Its representative masses, separations, temperatures, and times are design targets, not measured performance.

What a witness would establish

For a protocol tuned to $\Phi=\pi$, let $\lvert\Psi_{\mathrm{Bell}}\rangle$ be the locally phase-corrected target. One transparent witness is

$$\mathcal W_{\mathrm B}=\frac12 I-\lvert\Psi_{\mathrm{Bell}}\rangle\langle\Psi_{\mathrm{Bell}}\rvert.$$

Every separable two-qubit state satisfies $\operatorname{Tr}(\rho\mathcal W_{\mathrm B})\ge0$; a statistically significant negative value certifies entanglement. The target phase and local basis must be fixed by a preregistered gravitational model or an independent calibration. Optimizing them on the same noisy data changes the null distribution and requires a trials correction. Full spin tomography can instead report a simultaneous confidence region for $\rho$ and the maximum witness value over that region. A negative maximum certifies that every state still allowed by the confidence region is entangled. A negative minimum alone would establish only that some compatible state is entangled.

The mediator conclusion adds premises to the state conclusion. No direct quantum interaction may connect the masses; each local laboratory may interact only with the mediating gravitational system; and the latter must compose with the laboratories in the assumed way. Marletto and Vedral formulate the inference in terms of complementary mediator observables in Marletto and Vedral 2017, pp. 2–3. A later theorem allows continuous and infinite-dimensional matter systems but models a classical mediator by a commutative unital algebra and alternating local completely positive channels. Its initial joint state of both masses and the mediator must be triseparable, meaning a mixture or an appropriate limit of products across all three systems; independent preparation is sufficient. Under these hypotheses, and for a countable interaction sequence whose weak-* limit exists, the final two-mass state remains separable Ludescher et al. 2026, § III, corollary 1 and lemma 2, pp. 4–6. A separable reduced two-mass state alone does not exclude entanglement hidden in correlations with a mediator’s classical label.

There is a genuine adverse result. Aziz and Howl treat matter fully as a quantum field while leaving gravity classical and identify virtual-matter contributions that they argue can entangle the bodies through local processes; the predicted contribution has different parameter scaling from graviton exchange Aziz and Howl 2025, eqs. (8)–(10), pp. 815–817. This does not license the slogan that “any classical channel entangles.” It exposes a disputed decomposition: whether the quantum matter propagator is an additional quantum mediator, part of the gravitational interaction, or inconsistent with the stipulated local commutative mediator. A robust claim must therefore report the tested mass-and-time scaling and state exactly which degrees of freedom were excluded from the mediation channel.

Even under the strongest inference, the result would establish a low-energy noncommuting feature of gravity-mediated dynamics. It would not select loop, string, causal, or asymptotic-safety microphysics, nor directly probe Planckian curvature.

Evidence and dataset status through 2026-08-20

This discussion reflects results available through 2026-08-20. No qualifying public dataset reporting gravity-mediated entanglement between two separated masses was identified by this cutoff, and no such detection is treated here as established. The 2025 field review still presents gravity-mediated entanglement as a proposal and open experimental objective Bose et al. 2025, §IV.C, pp. 35–39; §VI, p. 56. Consequently there are no observed spin counts, fitted gravitational phase, apparatus data release, or experimental likelihood to reproduce on this page.

Record	Apparatus or result version	Public gravity-witness data at the cutoff
Bose et al. 2017	Spin-tagged two-mass Stern–Gerlach proposal; parameters in eqs. (1)–(7)	None; analytic sensitivity and background forecast only
Marletto–Vedral 2017	Two-path mediator theorem and interferometric proposal	None; conceptual protocol rather than completed apparatus
Bose et al. 2025	RMP survey of massive quantum systems, gravity proposals, and enabling control	No completed gravity-mediated-entanglement dataset reported
Aziz–Howl 2025 and Ludescher et al. 2026	Competing theoretical analyses of classical mediation	Not observational datasets; they alter the interpretation contract

This registry distinguishes enabling milestones—mass cooling, Stern–Gerlach recombination, force sensing, and nongravitational mechanical entanglement—from the joint gravity-only witness. None of those milestones alone supplies the missing dataset.

Preregistered analysis contract

Dataset and version. A future claim should identify the apparatus configuration, immutable raw-data release or repository DOI, acquisition dates, pulse sequence, geometry, and analysis commit. Proposal numbers must never be relabeled as observations.

Calibration. Each run needs traceable mass and mass-density estimates; branch separation and closest approach; interaction time; trap release and recombination transfer functions; magnetic-gradient and spin-rotation calibration; phase reference; readout confusion matrix; and covariance among these quantities. Calibration data must bracket the science run rather than be selected after seeing the witness.

Selections. Predeclare vacuum and temperature ranges, charge-neutralization acceptance, geometry tolerance, coherence threshold, pulse-quality and trap-loss cuts, excluded vibration or magnetic transients, heralding rules, and every post-selection. Report both the selected ensemble and all prepared trials so a success-conditioned witness cannot hide losses.

Likelihood, statistic, and priors. For settings $x,y$ and spin outcomes $a,b$, retain counts $n_{ab\mid xy}$ and use a multinomial likelihood

$$\mathcal L(\rho)= \prod_{x,y}\prod_{a,b} p(a,b\mid x,y,\rho)^{n_{ab\mid xy}},$$

including the calibrated readout channel. A frequentist analysis should give a confidence bound on $\operatorname{Tr}(\rho\mathcal W_{\mathrm B})$ under the separable null, with coverage checked by simulation. A Bayesian analysis must publish its state-space, nuisance, and phase priors and give prior-sensitivity results. Neither procedure may treat the phase or selected witness as fixed if it was chosen from the same data.

Systematics. The nuisance model must include residual charge and dipoles, patch potentials, Casimir and Casimir–Polder forces, magnetic cross-talk, common pulse fields, mechanical vibration, gravity gradients, gas collisions, thermal radiation, leakage between spin and motion, decoherence, readout bias, and classical communication through controls. Useful null runs reverse the branch geometry, detune the interaction time, block one superposition, vary separation, and use matched dummy masses. A force with the wrong $m_1m_2/d$ phase scaling is adverse evidence.

Reproduction status. Because no qualifying dataset was available, no empirical witness, uncertainty, or background-only fit was rerun here. The algebraic concurrence calculation below is a teaching check, not an experimental reproduction.

Exercise

Suppose local phases have been removed and the only nonzero branch phase is $\phi_{RL}=\pi/3$. Compute $\Phi$ and the ideal concurrence. Would the resulting entanglement, by itself, prove that the gravitational field is quantum?

Solution

Here

$$\Phi=0+0-0-\frac{\pi}{3}=-\frac{\pi}{3},$$

so

$$\mathcal C= \left\lvert\sin\left(-\frac{\pi}{6}\right)\right\rvert =\frac12.$$

The state is entangled in the ideal two-qubit model. An experiment must still establish a negative witness with calibrated uncertainty and exclude electromagnetic, mechanical, control, and selection-induced correlations. Inferring a nonclassical gravitational mediator additionally requires local mediation, an initially triseparable joint state of both masses and the mediator, no direct quantum channel, and a declared composition of matter with the mediator. Independent preparation of all three systems suffices for the initial-state condition; independence of the reduced two-mass state alone does not. The Aziz–Howl mechanism is an explicit reason to test the mass-and-time scaling rather than infer quantization from concurrence alone.

References

• Aziz, Joseph, and Richard Howl (2025), “Classical Theories of Gravity Produce Entanglement,” Nature 646, 813–817, doi:10.1038/s41586-025-09595-7.
• Bose, Sougato, et al. (2017), “Spin Entanglement Witness for Quantum Gravity,” Physical Review Letters 119, 240401, doi:10.1103/PhysRevLett.119.240401.
• Bose, Sougato, et al. (2025), “Massive Quantum Systems as Interfaces of Quantum Mechanics and Gravity,” Reviews of Modern Physics 97, 015003, doi:10.1103/RevModPhys.97.015003.
• Ludescher, Stefan L., Leon D. Loveridge, Thomas D. Galley, and Markus P. Müller (2026), “Gravity-Mediated Entanglement via Infinite-Dimensional Systems,” Journal of Physics A: Mathematical and Theoretical 59, 215302, doi:10.1088/1751-8121/ae65e7.
• Marletto, Chiara, and Vlatko 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.