Dear Antoine,
I very much enjoyed your essay (as I have previously enjoyed your papers and our discussions).
So, let me take the opportunity to ask for your reaction to my challenge to one particular aspect of your posture. It concerns an issue which we have previously discussed to some extent, but I think your essay gives us the opportunity to get a bit deeper into it .
I am referring to the robustness of the linearity requirement. In fact, I would like to challenge the generality of the following statement: `` The price is stiff: non-linearity is allowed at the wave-function level, but it has to vanish exctly at the density matrix level".
I know you have acknowledged explicitly that N. Gisin no-go theorem (as all no-go theorems) is only as strong as his assumptions (which include as far as I understand, no considerations of possible limitations on what can in fact be measured, other than those imposed by causality itself, as well as other possibilities that you raised yourself, like deviations from Born's Rule) .
What I want to consider next are some concrete reasons to doubt the strict validity of that linearity as a characteristic of viable quantum theories.
Let's start by assuming that some relativistic version of quantum theory involving spontaneous collapse is the adequate description of nature to a very good approximation at least in regimes where spacetime can be described by general relativity and the objects we are describing in quantum terms, are so small and light that their gravitational effects can be ignored, i.e. they are just test objects as far as gravity is concerned. If we depart substantially from the last condition there is of course good reasons to think that we would eventually enter a regime where a full quantum theory of gravity (QG) will be needed, and that quite possibly the standard notions of space-time would be lost. At that point of course our collapse theory will be meaningless as well, as it is formulated on the assumption that we have a suitable notion of time available to us (as in the nonrelativistic versions of spontaneous collapse theories are Schrödinger-like equations prescribing the time evolution of quantum states) or, a notion of spacetime (as used say in the relativistic versions such as Bedigham's , Tumulka's or Pearle's).
So let me consider, instead of jumping right into that QG regime, an idealized one parameter set of situations connecting the regimes where gravity might be ignored, passing trough one where it might suitably approximated by its Newtonian description ( as you have considered yourself), to one requiring that full QG regime where the vey notion of spacetime itself might be gone. Along that one parameter set of situations I expect we would, encounter situations where spacetime retains its usual meaning, but still, truly general relativistic effects will become relevant. As you know very well general relativity involves fundamental nonlinearities. Moreover, GR implies that the state of matter, by affecting the spacetime geometry itself does also, in general, affect its causal structure. Thus, it seems that there should be, along that one parameter set of situations, points where we have both: notions of spacetime (so might still be in the realms where one could sensibly use some version of spontaneous collapse theories) and, still some nonlinearities would start arising from the GR aspects of the problem.
That would, it seems to me, imply that at some point the linearity must be broken, that the superposition principle will have to give way to something else. The superposition principle would then survive as a good approximation valid in situations where gravity could be ignored, or at least where it could be treated within some linear approximation, such as a that provided by Newton's theory, or even linearized GR. In those situations the theory would indeed reduce to one satisfying linearity at the density matrix level. In more general situations it would not.
Now, how could something like this avoid being ruled out by Gisin's no-go theorem. One possibility is that the experimental arrangements envisioned in the theorem, (as they would in the pertinent case, certainly involve important gravitational effects) would be impossible to realize as a result of the modified theory itself. That is, the arrangements devised so that Alice could send a faster than light signal to Bob, might involve for instance, the setting up of some sort of superposition of energy momenta distributions, corresponding to spacetime superpositions, which according to the theory, would be simply impossible to achieve. On might imagine for instance that according to the theory, a collapse might have to take place, with probability 1, before Alice and Bob are able to complete the ensemble the experimental setup. In fact there is precedence for the impossibility of certain type of measurements (not involving gravitation) , that at first sight seemed quite feasible [ see for instance, Y Aharonov, D. Albert, ``Can we make sense out of the measurement process in relativistic quantum mechanics?" PRD 24, 359 (1981) & R. Sorkin, ``Impossible measurements on quantum fields", in Directions in general relativity: Proceedings of the 1993 International Symposium, Maryland, Vol. 2, 293 (1993)]. Another possibility, taking us a bit outside what I had been considering, is that the attempt to create the set-up, would involve, in a sense, creating something like a "spacetime causal structure which is not well defined", so that at the end, whether or not a signal was sent faster than light between Alice and Bob would remain undecidable. Actually things might turn out differently and the causal structure might end up being emergent, and defined depending on the ``outcome" of the experimental development itself. Something of this nature is exemplified in [ "Large fluctuations in the horizon and what they teach us about Quantum Gravity and Entropy" R. Sorkin y D. S., CQG16, 3835, (1999); When is S =1/4 A ? ", A. Corichi & D. S. ,MPLA 17, pg. 1431, (2002), and "A Schrodinger Black Hole and its Entropy", D. S., MPLA17, 1047, (2002)]. Another interesting option is that in the context at hand ( and as a result of the fact that in the semi-classical description one would be ignoring the quantum nature of the gravitational degrees of freedom), Born's rule would end up being effectively modified as you suggested yourself ( but in what I took you considered as highly improbable development, please correct me if I read it wrongly). In other words to expect the unexpected does not seem out of place in dealing with the interface of gravitation and quantum theory.
In fact, it seems to me that several of the steeps used in the derivations presented in the essay, in particular those that involve taking averages over ensembles, ought to be revised in the kind of gravitational context I am describing, for various reasons. To start with we do not even know how to make sense of the sum of two spacetimes, and much less "the average of various spacetimes", and even if we did, it seems quite likely that the intrinsic nonlinearities of GR would invalidate some of the usual steeps averaging procedures.
It is of course a tall order to actually propose such a theory, but accepting that we might have to consider breaking with linearity (even at the level of the density matrix equation) might be the first necessary steep.
I clearly do not expect you (or anybody at this time) to have any clear and definite answers to the questions raised by the above considerations ( I might be wrong of course) , but I am just curious to see what your first reaction would be.
Again, congratulations for a very nice essay (which I will acknowledge with a very high mark) .
