Minimal noise in non-quantized gravity

Abstract: 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.

• The paper introduces a unified framework for non-quantized gravity by parametrizing gravitational noise with a Lindblad master equation.
• It derives minimal noise thresholds and entanglement criteria with explicit numerical bounds, setting clear experimental targets.
• The analysis contrasts classical-quantum and entropic gravity models, revealing how irreversibility and entanglement can coexist.

Minimal Noise in Non-Quantized Gravity: A Technical Assessment

Formal Framework for Non-Quantized Gravity Models

This paper systematically characterizes models in which gravity is not quantized, providing a unified framework to describe their dynamics in the non-relativistic regime. The evolution of massive objects is constructed under three minimal assumptions: Galilean invariance, time locality, and averaged Newtonian evolution. The general time evolution for pairs of masses is formalized as a Lindblad master equation, encompassing both entangling Hamiltonian terms and dissipation representing irreversible, noisy evolution. The central result is an explicit parametrization of the most general forms of gravitational noise consistent with these constraints, via rotationally invariant functions governing Lindblad operators over the center-of-mass and relative coordinates.

The dissipative part of the evolution takes the form

$$\mathcal{D}(\rho) = \sum_i \int d^3 k \Big[e^{i k \cdot R} J_{k,i}(r) \rho J_{k,i}^\dagger(r) e^{-i k \cdot R} - \frac{1}{2}\{J_{k,i}^\dagger(r) J_{k,i}(r), \rho\}\Big]$$

where all model-specific structure is in $J_{k,i}(r)$. This form includes both classical-quantum and measurement-feedback models. In quantum gravity, $J_{k,i} \equiv 0$; all non-zero dissipation terms encapsulate deviations from graviton physics, signifying irreversibility and decoherence.

Figure 1: Two mechanical oscillators separated by $d$, representative of current tabletop experiments probing gravitational interactions.

Quantitative Noise Bounds and Entanglement Thresholds

The examination of noise is formalized via excess rates of physically measurable observables (such as momentum variance or phase coherence) induced by gravitational dissipation. For two mechanical masses, the force noise spectral density is directly derivable from the Lindblad parametrization and can be operationally compared to experimental sensitivity.

A key theoretical result is the existence of a minimal, quantifiable noise threshold for any non-entangling model:

$S_{FF,e} < \frac{4 G_N m^2}{d^3}$

for pairwise noise rates and

$\Gamma_e < \alpha_G \delta x^2$

for superposed two-state masses. If observed noise is below these bounds, Newtonian gravity must be entangling. The numerical levels specified are not astronomically smaller than state-of-the-art force measurements; for $d = 1$ mm, noise thresholds are $\approx 10^{-18}~{\rm m/s^2}/\sqrt{\rm Hz}$, roughly three orders below LISA Pathfinder sensitivity.

The analysis is extended across multiple experimental architectures: pairs of mechanical oscillators, two masses in positional superpositions, and hybrid oscillator-qubit setups, demonstrating that each system yields sharply defined noise/entanglement boundaries and practical experimental targets.

Application to Classical-Quantum and Entropic Gravity Models

Concrete models are positioned within the general framework, notably classical-quantum (CQ) gravity and entropic gravity proposals.

CQ gravity is defined by stochastic classical fields coupled to quantum matter, with two central parameters: diffusion ($D_2$) and decoherence ($D_0$), bound via $D_0 D_2 \geq 1$. Force noise in CQ theory is dictated by these parameters, setting a lower bound

$$S_{FF,e} \geq \frac{8 \sqrt{5} G_N m^2}{15 \pi \ell^3} \gtrsim \frac{4 G_N m^2}{d^3}$$

As experiments approach noise sensitivities below this bound, increasingly larger portions of CQ parameter space are excluded. Notably, CQ models cannot be arbitrarily less noisy without violating self-consistency, and thus cannot evade experimental falsification simply by parameter tuning.

Entropic gravity models, whether local or non-local, are scrutinized for their noise and entanglement properties. The non-local entropic model, which features mediator degrees of freedom coupling via relative positions, demonstrates that gravity can be both noisy and entangling—contrary to common expectations regarding irreversibility forbidding entanglement. The local entropic model, in contrast, acts as a noise source degrading entanglement, and is excluded if gravitational experiments detect entanglement or if the noise is observed below theoretical bounds.

Figure 2: Parameter space of classical-quantum gravity, showing constraints from tradeoff relations and current/future noise measurements.

Implications for Quantum Gravity Experiments

The presented framework yields practical, quantitative targets for ongoing gravitational experiments. The correlation between entanglement generation and minimal noise rates implies that measuring noise below the derived thresholds is logically equivalent to proving the quantization of gravity in the sense of entangling interactions. Conversely, failing to observe entanglement or failing to measure excess noise provides systematic constraints on non-graviton models. The framework uniquely accommodates non-Gaussian and correlated noise, crucial for accurately representing specific gravitational alternatives, such as CQ and entropic gravity.

Importantly, the results illustrate that theoretical predictions regarding the quantum versus classical nature of gravity can be tested incrementally—experimentally excluding vast regions of parameter space before reaching ultimate thresholds. This sets the stage for progressive refinement of gravitational theories vis-à-vis empirical data, enhancing the interplay between fundamental physics and precision measurement.

Conclusion

The paper constructs a rigorous parametrization of general gravitational models, demonstrating that any non-quantized, non-entangling framework capable of recovering Newtonian gravity on average must inject a calculable, irreducible amount of noise. The implications for experimental quantum gravity are direct and actionable: both noise and entanglement generation are necessary and sufficient proxies for graviton physics. The technical structure here provides a basis for constraining or falsifying alternative models as experimental sensitivity advances. The framework also clarifies scenarios where entangling interactions coexist with irreversibility, emphasizing the need for multi-faceted experimental approaches.