Assume that it is possible to prepare a macroscopic system (say a 1kg iron sphere) in a superposition of two position eigenstates 1 meter apart. This experiment has to be isolated from the environment, so let's assume that we place it inside a magical box that does not let any particle disturb the system. Outside the box we place a torsion balance. As far as I can tell the following premises are true:

  1. No matter how the "magical box" is built the gravitational field of the iron sphere will still be detectable by the torsion balance.

  2. The torsion balance actualy measures the object's position because the gravitational field points to the object's center of mass.

  3. Once the position of the iron sphere is known, its state "collapses" into one of the two eigenstates.

  4. Any massive object can play the role of the torsion balance.

From 1,2,3 and 4 it follows that such a superposition state is incompatible with the presence of any massive object outside the box. But in order for this experiment to make sense you need at least one object outside the box, the observer. So, it appears to me that such superpositions cannot exist.

Note: it is always possible to have a superposition as long as the measurement error implied by the uncertainty principle is enough to "protect" it. So the above argument is compatible with experiments such as the two-slit experiment. For the proposed experiment with the iron sphere the uncertainty principle is not relevant.

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    $\begingroup$ Experiments to explore this have been proposed. A recent example is Testing the gravitational field generated by a quantum superposition, and a recent review is Tabletop experiments for quantum gravity. $\endgroup$ – Chiral Anomaly Jul 19 at 12:58
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    $\begingroup$ I know about these proposed experiments but it seems to me that they contradict QM. It should not be possible to measure an object as a superposition (they expect to see superposed gravitational fields originating from the two superposed locations). Let's assume we can measure the gravitational field from both locations. What if we shine some light on the objects and perform a position measurement this way? Would the gravitational field "collapse" to one location at that moment? $\endgroup$ – Andrei Jul 19 at 13:48
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    $\begingroup$ We can do interference experiments with electrons even though an electron has an electric field, and its electric field influences other charged particles more strongly than its gravitational field does. Of course, the proposed experiments are not using anything close to 1 kg of iron or locations separated by 1 meter. Observing interference in those conditions is probably impossible. Determining exactly how large the mass and separation must be to make interference experiments impossible would require thinking about what "impossible" means, which I think is what makes the question interesting. $\endgroup$ – Chiral Anomaly Jul 19 at 23:52
  • $\begingroup$ In the case of electrons we need to take into account the uncertainty principle. detecting the position of the electron with enough precision to get "which-slit"information will make the interference pattern disappear. Using the electric field or the gravitational field should make no difference because the back reaction is the same. For macroscopic superpositions such as the case of the iron balls the uncertainty principle will not be significant. $\endgroup$ – Andrei Jul 22 at 4:02

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