Centre of mass decoherence due to time dilation: paradoxical frame-dependence

The recently proposed centre-of-mass decoherence of composite objects due to gravitational time-dilation [Pikovski et al, Nat.Phys. 11, 668 (2015)] is confronted with the principle of equivalence between gravity and observer's acceleration. In the laboratory frame, a positional superposition |x1〉 + |x2〉 can quickly decohere whereas in the free-falling frame, as I argue, the superposition can survive for almost arbitrary long times. The paradoxical result is explained by the so far unappreciated feature of the proposed model: the centre-of-mass canonical subsystem is ambiguous, it is di˙erent in the laboratory and the free-falling frames, respectively. As long as the centre-of-mass motion of the composite object is non-relativistic, a simple Galilean-covariant Hamiltonian represents the Pikovski et al theory with exactly the same physical predictions. We emphasize the power of this Hamiltonian to understand essential features of the Pikovski et al theory and to moderate a few divergent statements in recent works.