What are the strongest objections to be made against decoherence as an explanation of "collapse?"

I think most arguments in the literature can be boiled down to the point that decoherence does in no way touch the linearity of the Schrödinger equation, and thus cannot make an "or" from an "and". This is complicated in the literature by very technical discussions, which I would like to avoid.

Let me explain the basic point in more details. A widely referenced statement of the measurement problem is given in the paper "Three measurement problems" by Tim Maudlin, section 1 (first two pages, whole paper). He considers a spin measurement apparatus, which has the property that if you put in an electron in spin+ eigenstate, a pointer points to the left, and if the electron comes in with a spin- eigenstate, a pointer points to the right. Now it follows simply from the linearity of the Schrödinger evolution that the initial state $$\rm\frac{1}{\sqrt{2}} \left(\left|up\right>_e + \left|down\right>_e\right) \otimes \left|ready\right>_d $$ evolves to the final state, after measuring, $$\rm\frac{1}{\sqrt{2}} \left(\left|up\right>_e \otimes \left|LEFT\right>_d + \left|down\right>_e \otimes \left|RIGHT\right>_d\right). $$ This looks like a pointer pointing to the left and to the right, which cannot be true and is not what we see in labs or around us. We somehow have to make an "or" out of this "and to get the correct facts.

Among people who believe that this is a problem and that decoherence doesn't solve it, the prevailing point of view seems to be the one given in a letter by Adler, which also references other literature. To solve this problem, which is in fact the so-called measurement problem, there are several options now:

• change the Schrödinger equation (done in so-called collapse models)
• add additional local beables (in Bell's words)/ things other than the wave function to the theory, as done in de-Broglie-Bohm theory
• postulate that measurements actually don't have unique outcomes and we somehow only perceive single outcomes when we do experiments (the many-worlds-path).

For many practical purposes, the collapse postulate of the Copenhagen interpretation works fine. Now if you look for an explanation of the collapse, one of the three options above would serve as one.

Decoherence can only explain why the collapse works well: Because the wave function parts that are disregarded won't interfere again with the ones we keep, the collapse is fine. So decoherence explains why you can, in many worlds, separate the worlds and say they do not come in contact again. In de-Broglie-Bohm, it explains why you can effectively collapse the wave function after a measurement. But the collapse itself is not explained, there is still the sum in the above formula, still an "and", and how to get a definite result from it can only go along one of the three lines I sketched.

The simplest way to phrase the main objection is that decoherence doesn't even try to solve anything beyond "for all practical purposes." It is a way of describing entanglement with the environment that is irreversible "in practice," but what may be impractical today may well be practical tomorrow (or for a more advanced civilization even today). As indicated elsewhere, the evolution is still understood to be unitary in principle.

Roger Penrose (2004), The Road to Reality, pp. 802-803:

...the environmental-decoherence viewpoint ... maintains that state vector reduction [the R process] can be understood as coming about because the environmental system under consideration becomes inextricably entangled with its environment.[...] We think of the environment as extremely complicated and essentially 'random' [...] Under normal circumstances, one must regard the density matrix as some kind of approximation to the whole quantum truth. For there is no general principle providing an absolute bar to extracting information from the environment.[...] Accordingly, such descriptions are referred to as FAPP [For All Practical Purposes]

Maybe a future technology could provide means whereby quantum phase relations can be monitored in detail, under circumstances where present-day technology would simply ‘give up’. It would seem that the resort to a density-matrix description is a technology-dependent prescription! With better technology, the state-vector description could be maintained for longer, and the resort to a density matrix put off until things get really hopelessly messy!

Addendum

There are also attempts at demonstrating decoherence that is irreversible in principle. See related question: Can the Montevideo interpretation of quantum mechanics do what it claims?