To the Editor

The interface between gravitation and quantum theory is a fascinating subject. However, it is also riddled with subtleties, and slight confusion can easily lead to questionable conclusions. A striking example in this regard is provided by the work of Pikovski et al.1, in which it is claimed that gravitational effects generically produce a novel form of decoherence for systems with internal degrees of freedom, which would account for the emergence of classicality. The effect is supposed to arise from the different gravitational redshifts suffered by such systems when placed in superpositions of positions along the direction of the gravitational field. There are, however, serious issues with the arguments of the paper.

First, the results of ref. 1 cannot be right in light of the equivalence principle, which is valid, by construction, in the frameworks used. This is because the only external force acting on all studied systems is generated by a gravitational field, and no spacetime curvature effects are relevant. As a result, the situations analysed are equivalent to ones without gravity, in which an accelerated observer studies free, isolated systems. Clearly, such scenarios cannot lead to decoherence as, without gravity, there is nothing to cause it. Moreover, as the systems described in ref. 1 are subject to gravity, they will not remain static when placed in a superposition of fixed positions. Of course, one could achieve this by including a compensating force generated by an external device, and this additional interaction may lead to decoherence, but this effect cannot be ascribed to gravity.

Next, notice that a central premise of the analysis presented in ref. 1 asserts that the system's internal energy contributes to its effective mass, which thus can have more than one value. However, ordinary non-relativistic quantum mechanics cannot deal with such situations2, a fact ignored by Pikovski et al.1 in their free use of such a framework.

Finally, the widespread belief that decoherence can explain the quantum-to-classical transition, which is key in the analysis of the paper, is unjustified3. The confusion arises from the fact that the density matrix of an improper mixture (which represents the partial description of a subsystem that is part of a larger system in a pure state) has, after decoherence takes place, the same form as that of a proper mixture (which represents an actual ensemble of systems)4. It does not, however, follow from such formal similarity that the two physical situations are identical. Therefore, even if the reduced density matrix for the centre of mass of the systems considered in ref. 1 has the same form as a statistical mixture, it does not follow that their physical situation is indistinguishable from that of an ensemble; the centre of mass continues to be as entangled and delocalized as it was before the alleged decoherence took place.

We conclude, from all this, that the claims of Pikovski et al. in ref. 1 are invalid. An in-depth analysis of the results and claims in ref. 1 is provided in ref. 5.