Last week two curious papers appeared on the arXiv, one by Marletto and Vedral, and the other by Bose et al., proposing to test whether the gravitational field must be quantized. I think they have a nice idea there, that is a bit obscured by all the details they put in the papers, so I hope the authors will forgive me for butchering their argument down to the barest of the bones.

The starting point is a worryingly common idea that maybe the reason why a quantum theory of gravity is so damn difficult to make is because gravity is not actually quantum. While concrete models of “non-quantum gravity” tend to be pathological or show spectacular disagreement with experiment, there is still a lingering hope that somehow a non-quantum theory of gravity will be made to work, or that at least a semi-classical model like QFT in a curved spacetime will be enough to explain all the experimental results we’ll ever get. Marletto and Bose’s answer? Kill it with fire.

Their idea is to put two massive particles (like neutrons) side-by-side in two Mach-Zender interferometers, in such a way that their gravitational interaction is non-negligible in only one of the combination of arms, and measure the resulting entanglement as proof of the quantumness of the interaction.

More precisely, the particles start in the state \[ \ket{L}\ket{L}, \] which after the first beam splitter in each of the interferometers gets mapped to \[ \frac{\ket{L} + \ket{R}}{\sqrt2}\frac{\ket{L} + \ket{R}}{\sqrt2} = \frac12(\ket{LL} + \ket{LR} + \ket{RL} + \ket{RR}), \] which is where the magic happens: we can put these interferometers together in such a way that the right arm of the first interferometer is very close to the left arm of the second interferometer, and all the other arms are far away from each other. If the basic rules of quantum mechanics apply to gravitational interactions, this should give a phase shift corresponding to the gravitational potential energy to the $\ket{LR}$ member of the superposition, resulting in the state

\[ \frac12(\ket{LL} + e^{i\phi}\ket{LR} + \ket{RL} + \ket{RR}), \] which can even be made maximally entangled if we manage to make $\phi = \pi$. If we now complete the interferometers with a second beam splitter, we can do complete tomography of this state, and in particular measure its entanglement.

Now I’m not sure about what “non-quantum gravity” can do, but if it can allow superpositions of masses to get entangled via gravitational interactions, the “non-quantum” part of its name is as appropriate as the “Democratic” in Democratic People’s Republic of Korea.