A no-go theorem on the nature of the gravitational field beyond quantum theory

Recently, table-top experiments involving massive quantum systems have been proposed to test the interface of quantum theory and gravity. In particular, the crucial point of the debate is whether it is possible to conclude anything on the quantum nature of the gravitational field, provided that two quantum systems become entangled due to solely the gravitational interaction. Typically, this question has been addressed by assuming an underlying physical theory to describe the gravitational interaction, but no systematic approach to characterise the set of possible gravitational theories which are compatible with the observation of entanglement has been proposed. Here, we introduce the framework of Generalised Probabilistic Theories (GPTs) to the study of the nature of the gravitational field. This framework has the advantage that it only relies on the set of operationally accessible states, transformations, and measurements, without presupposing an underlying theory. Hence, it provides a framework to systematically study all theories compatible with the detection of entanglement generated via the gravitational interaction between two non-classical systems. Assuming that such gravitationally mediated entanglement is observed we prove a no-go theorem stating that gravity cannot simultaneously satisfy the following conditions i) it is a theory with no faster-than-light signalling; ii) it mediates the gravitational interaction via a physical degree of freedom; iii) it is classical. We further show what the violation of each condition implies, and in particular, when iii) is violated, we provide examples of non-classical and non-quantum theories which are logically consistent with the other conditions.