The question of whether gravity is quantum remains one of the central problems in modern physics. The proposal of Bose et al. (2017) showed that if two massive particles become entangled through the gravitational interaction alone, then the gravitational field cannot be classical. However, the mechanism behind the entanglement generation, that is, exactly how gravity redistributes quantum resources, remained unclear. This work fills that gap by treating the gravitational interaction from the standpoint of quantum resource theory. It belongs to a line of research that probes the quantum aspects of gravity through the entanglement of masses, but it stands out by focusing on an exact quantitative description of how resources are transformed.
The authors use a model of two neutral test masses, each prepared in a spatial superposition of two localized states |L⟩ and |R⟩. The gravitational interaction imprints different phases on the different branches of the superposition, which is described by a unitary transformation. The analysis relies on quantum resource theory: coherence is measured by the l1 norm and by relative entropy, entanglement by negativity and entanglement entropy. Analytical expressions for these quantities are derived as functions of the accumulated phase, and generalized initial states with arbitrary amplitudes are also studied numerically.
The gravitational interaction is shown to be a unitary incoherent operation that preserves the total coherence of the system. For an initially maximally coherent state, exact complementarity relations are obtained: the square of the l1 norm of local coherence plus the square of negativity equals one, and the sum of the relative entropy of coherence and the entanglement entropy also equals one. This means that local coherence and entanglement are inversely related: as one grows, the other shrinks. For arbitrary initial states, inequalities are derived that become equalities only at maximal initial coherence of both subsystems. The key conclusion: if even one subsystem starts out incoherent, no entanglement arises at all.
The work sharpens the interpretation of experiments that test quantum gravity: to observe gravitationally induced entanglement successfully, one must prepare and maintain sufficient quantum coherence in the test masses. This requirement becomes an explicit criterion for the design of future experiments. Unlike neighbouring studies, which explore whether masses can become entangled or how decoherence interferes, this paper is the first to tie initial coherence quantitatively and exactly to the resulting entanglement within a specific model, using the machinery of quantum resource theory.
A natural next step is to include realistic sources of decoherence and imperfections in state preparation, in order to estimate how close real experiments can come to the theoretical entanglement limit. It would also be interesting to extend the analysis to more general gravitational interactions, including relativistic corrections and many-particle systems.
The results will shape the planning of experiments that test the quantum nature of gravity, since they set requirements on the initial coherence of the masses. They also enrich quantum resource theory with an example of a physically motivated channel for resource conversion.
The next step could be an analysis of how robust the complementarity relations are against noise and losses, along with a search for optimal state-preparation protocols that maximize entanglement under limited resources.
The work is directly tied to the fundamental problem of quantum gravity: is the gravitational field quantum? It refines the test criterion based on the entanglement of masses by showing that the quantum coherence of the initial states is a necessary condition. This links the problem to a broader set of questions about the role of quantum resources in physical interactions.
Maximal entanglement between two masses is reached exactly when the local coherence of each mass vanishes completely, so gravity, as it were, pumps the quantum resource out of each particle and into their joint state.