Minimal noise in non-quantized gravity

An elementary prediction of the quantization of the gravitational field is that the Newtonian interaction can entangle pairs of massive objects. Conversely, in models of gravity in which the field is not quantized, the gravitational interaction necessarily comes with some level of noise, i.e., non-reversibility. Here, we give a systematic classification of all possible such models consistent with the basic requirements that the non-relativistic limit is Galilean invariant and reproduces the Newtonian interaction on average. We demonstrate that for any such model to be non-entangling, a quantifiable, minimal amount of noise must be injected into any experimental system. Thus, measuring gravitating systems at noise levels below this threshold would be equivalent to demonstrating that Newtonian gravity is entangling. As concrete examples, we analyze our general predictions in a number of experimental setups, and test it on the classical-quantum gravity models of Oppenheim et al., as well as on a recent model of Newtonian gravity as an entropic force.

💡 Research Summary

The paper addresses a fundamental question in the ongoing debate over whether gravity must be quantized: if gravity is not mediated by quantum gravitons, does the Newtonian interaction inevitably introduce irreversibility in the form of noise? The authors answer affirmatively and provide a systematic framework that captures every conceivable non‑quantized gravity model consistent with three minimal physical assumptions: (i) the average dynamics obey Newton’s law (Ehrenfest condition), (ii) the evolution is time‑local, and (iii) the dynamics respect Galilean symmetry.

Starting from these premises, they derive the most general Lindblad master equation for two non‑relativistic massive particles interacting gravitationally. The Hamiltonian part contains the usual kinetic energy and the Newtonian potential, while the dissipative part is expressed through a set of Lindblad operators (L_{\lambda}=e_i\cdot R,J_{k,i}(r)). Here (r=x_1-x_2) is the relative coordinate, (R) the centre‑of‑mass coordinate, and (J_{k,i}(r)) are arbitrary rotationally invariant functions that encode the specific model. In a fully quantized theory (gravitons) all (J_{k,i}) vanish, leaving a purely unitary evolution; any non‑zero (J_{k,i}) signals a deviation from standard quantum gravity and introduces irreversible noise.

To make the formalism experimentally tractable, the authors present a simplified version of the master equation (Eq. 8) that isolates three physically meaningful contributions: (a) single‑particle decoherence described by isotropic functions (f_a(k)), which can be non‑Gaussian; (b) a non‑local correlated noise term proportional to a real coefficient (\beta) that acts jointly on both masses; and (c) the usual Newtonian entangling interaction expanded to leading order in the separation. This parametrization requires only a handful of scalar functions and a single coefficient, yet it is sufficient to reproduce all known non‑quantized gravity proposals, including the “classical‑quantum” models of Oppenheim et al. and the emergent entropic‑gravity model.

The central result is the identification of a minimal noise threshold. By defining appropriate noise observables (e.g., excess momentum variance for mechanical oscillators) and entanglement witnesses (Simon’s partial‑transpose criterion for continuous variables, or coherence decay for two‑level systems), the authors show that if the single‑particle noise strength falls below a certain bound (\Gamma_{\text{thresh}}), the Newtonian interaction must generate entanglement. Conversely, any model that fails to produce entanglement must inject at least this amount of noise. The threshold depends on the masses, separation, and oscillator frequencies, but for a prototypical setup (two millimetre‑separated micro‑gram masses) it corresponds to a force‑noise spectral density of order (\sqrt{S_{FF}}\sim10^{-18},\text{m,s}^{-2},\text{Hz}^{-1/2}), roughly three orders of magnitude below the LISA Pathfinder sensitivity.

The paper then applies the framework to two concrete models. In the “classical‑quantum” hybrid, the correlated noise (\beta) and the single‑particle spectra (f_a(k)) are derived from the underlying measurement‑feedback dynamics; the analysis reveals that non‑Gaussian single‑particle terms are essential and were missed in earlier Gaussian approximations. In the entropic‑gravity scenario, all decoherence originates from the correlated (\beta) term, leading to a different scaling of the noise bound but still respecting the same minimal‑noise principle.

From an experimental standpoint, the authors emphasize that any measured total noise (\Gamma_{\text{meas}}) provides an upper bound on the gravitationally induced noise (\Gamma_{\text{grav}}\le\Gamma_{\text{meas}}). Therefore, achieving a measured noise level below the derived (\Gamma_{\text{thresh}}) would constitute a direct demonstration that Newtonian gravity is capable of entangling masses, effectively ruling out the entire class of non‑quantized models that respect the three foundational assumptions.

In summary, the work extends the Kafri‑Taylor decoherence‑entanglement trade‑off to a fully general, non‑Gaussian, and potentially non‑local setting. It delivers a clear, quantitative criterion—minimum unavoidable noise—for any non‑quantized gravity theory, and translates this criterion into concrete experimental targets for tabletop optomechanics, atom interferometry, and hybrid spin‑mechanical platforms. By doing so, it bridges the gap between abstract theoretical proposals and realistic laboratory tests, offering a roadmap for the next generation of experiments that aim to settle the quantum versus classical nature of gravity.