Dear Daniel,
Thank you very much for your thoughtful comment on my essay, it means a lot.
I have been thinking quite a lot on this problem of linearity, and so before trying to answer your points on this, let me try to explain how I evolved on this question. As you have perhaps noted, my discussion of this in the essay is at the same time more careful and shorter than in previous articles.
At first, I encountered this need for linearity at the density matrix level when we were constructing semiclassical theories of gravity with Lajos Diosi. There, the main interest of our approach was precisely to remove non-linearity. And so to me, the achievement felt even more so amazing that I was convinced all non-linear approaches were doomed. I still think it is great to be able to remove non-linearity, and that having a theory that is linear at the master equation level is much more convenient, everything else being equal.
But at the same time I realized, and this is the core of the present essay, that asking for linearity, although it makes everything simpler, does not allow us to go beyond orthodox quantum theory at least at the empirical level (namely, all the experimental results are reproducible my quantum theory, only not the standard model). And so, by asking for a simplification, linearity, we remove at the same time anything genuinely new that could have happened empirically. Naturally, metaphysical interests remain (the measurement problem is solved), but the proposal becomes much less radical than one might have hoped.
And so, while I think the issues of non-linearity should not be underestimated, I think it is important to see also how requiring linearity removes the empirical novelty of these collapse like approaches.
Now, let me explain why I remain convinced that the price to pay to have fundamental non-linearity is much higher than people think. Nicolas Gisin's formulation of the no-go in terms of faster than light signaling is probably the most impressive, but in the end I do not think faster than light signaling is the main issue. In essence, I believe the problem is more one of predictability, and ability to separate systems into subsystems for all practical purposes. With non-linearity, the statistics of a subsystem is influenced by what happens arbitrarily far way, and so effectively we have a force without limit on its range. Further, while non-linearity can start very small, there is no reason to expect that it typically remains small macroscopicaly. Just think of the collapse process in GRW: very small but massive modifications for macroscopic bodies (because of linearity however, these massive modifications have no empirical signature). In general, the weirdness coming from non-linearity has no reason to be confined to microscopic degrees of freedom, unless there are precisely the right cancellations. If non-linearity comes from gravity, you can expect the macroscopic non-linear corrections to be of the same order of magnitude as gravity itself, hence dozens of orders of magnitude larger than quantum mechanical effects for macroscopic bodies.
How would we see these brutal modifications, or convince ourselves that they don't exist? It's very hard: with non-linear dynamics, the Born rule is no longer valid (a priori not even approximately valid). So you can't trust the wavefunction, you have to go back at the local beable level. Non-linearity also forbids the separation into subsystems. So you would have to prove that some sort of approximate Born rule can be derived for appropriate local beables, and that though systems cannot be exactly separated into subsystems even if they are far away (decoherence is not enough), you can still do it in most cases (because of a subtler non-linear decoherence). Usually, in approaches based on local beables like Bohmian mechanics, you have a rather straightforward argument (e.g. equivariance) to interpret the wavefunction probabilistically, but in a non-linear theory you have to do it from scratch and the best you can hope is that it holds approximately. Frank Laloë has tried to do this in a recent paper, where he tries to see that that the equivariant distribution is stable with respect to a small non-linear perturbation, at least for large bodies in the model he introduced. At my current level of understanding, I don't think that it works, but it is certainly an attempt in this direction.
So this was to insist that non-linearity makes things difficult, because all the tools we use to make predictions break down (and it's not clear they only approximately break down, because the violations may be huge for a measurement apparatus). I would find fundamental non-linearity interesting, because it would mean we could falsify quantum theory itself and not just its specific quantum field theory instance. But is there a reason that non-linearity has to exist because of quantum gravity for example? There, I am less convinced than you.
Quantum gravity is hard to discuss because it is not precisely defined yet. But if quantum gravity can ultimately be defined as something that looks like a slightly weirder quantum field theory (as String Theory aims to do for example), then the non-linearity of field equations has no reason to be translated into a non-linear dynamics on Hilbert space. The (quantum) self interacting scalar field has a non-linear field equation, yet its dynamics is purely unitary and can be rigorously constructed (at least in 1+1 and 2+1 dimensions). I don't understand why the non-linearity of gravity is different, for example, from the non-linearity of non-Abelian Yang-Mills.
Another option would be that quantum gravity brings weird causal superpositions, and situations where there is some form of faster than light signaling, or something hard to interpret. But this would not be related to non-linearity in the sense we discussed, and in this context I don't really worry about faster than light signaling (because indeed, it probably does not yield anything that can be exploited, maybe the observables can't be measured, and the effect does not grow into unacceptable macroscopic corrections because of decoherence). Again it's not so much an argument for non-linearity than an argument to say that some of the mildest consequences of non-linearity are acceptable since they could appear somewhere else.
Now in the essay, I merely want to state why we came to accept linearity, I think for good reasons (even though not necessarily with watertight arguments), but explain what the less appreciated empirical consequences are. Non-linearity makes things too difficult, linearity makes things simple but almost unique.
Again, many thanks for your comments, they had me thinking quite a lot. I hope we get a chance to discuss more in the future.
All the best,
Antoine