Einstein's Equivalence principle for superpositions of gravitational fields

The Principle of Equivalence, stating that all laws of physics take their special-relativistic form in any local inertial frame, lies at the core of General Relativity. Because of its fundamental status, this principle could be a very powerful guide in formulating physical laws at regimes where both gravitational and quantum effects are relevant. However, its formulation implicitly presupposes that reference frames are abstracted from classical systems (rods and clocks) and that the spacetime background is well defined. It is unclear if it continues to hold when quantum systems, which can be in a quantum relationship with other physical systems, are taken as reference frames, and in a superposition of classical spacetime structures. Here, we tackle both questions by introducing a relational formalism to describe quantum systems in a superposition of curved spacetimes. We build a unitary transformation to the quantum reference frame (QRF) of a quantum system in curved spacetime, and in a superposition thereof. In both cases, a QRF can be found such that the metric looks locally minkowskian. Hence, one cannot distinguish, with a local measurement, if the spacetime is flat or curved, or in a superposition of such spacetimes. This transformation identifies a Quantum Local Inertial Frame. We also find a spacetime path-integral encoding the dynamics of a quantum particle in spacetime and show that the state of a freely falling particle can be expressed as an infinite sum of all possible classical geodesics. We then build the QRF transformation to the Fermi normal coordinates of such freely falling quantum particle and show that the metric is locally minkowskian. These results extend the Principle of Equivalence to QRFs in a superposition of gravitational fields. Verifying this principle may pave a fruitful path to establishing solid conceptual grounds for a future theory of quantum gravity.