It seems there is a new theorem that changes the rules of the game in the interpretational debate on QM:
http://www.nature.com/news/quantum-theorem-shakes-foundations-1.9392
Does this only leave Bohm,Everett and GRW as possible candidates?
Answer
The paper does not go into details about what interpretations would be disproved by their results. There's a good reason for this: There are no interpretations that would be disproved by their results. They are disproving a straw-man. Here is the central result proved by the paper, phrased in a less obscure way:
"If a system is in the pure state $|+_Z\rangle$ then it is definitely not in some other different pure state $|+_X\rangle$ or whatever."
If this seems obvious and uncontroversial, it is! Admittedly, in the conclusions section, they claim they are saying things that are not obvious...but they're wrong.
Let's start from the beginning. They define the debate by saying there are two pure quantum states, $|\phi_0\rangle$ and $|\phi_1\rangle$. There is one procedure to prepare $|\phi_0\rangle$ and a different procedure to prepare $|\phi_1\rangle$. They say there are two schools of thought. The first school of thought (the correct one) is that "the quantum state is a physical property of the system", so that "the quantum state is uniquely determined by [the physical situation]". That's the one that they will prove is correct. They say the alternative (the incorrect one) is that "the quantum state is statistical in nature", by which they mean "a full specification of [the physical situation] need not determine the quantum state uniquely".
Let's say you have a spin-1/2 system, in state $|+_Z\rangle$. Then...HOLD ON A MINUTE! I just committed myself to the first school of thought! I said the system was really in a certain quantum state!
In fact, everyone doing quantum mechanics is always in the first school of thought, because we say that a system has a quantum state and we do calculations on how the state evolves, etc., if the system is in a pure state. (Not necessarily true for mixed states, as discussed below.)
What would be the second school of thought? You would say, "I went through a procedure which supposedly prepares the system into the pure state $|+_Z\rangle$. But really the system doesn't just have one unique state. It has some probability somehow associated with it. This exact same procedure might well have prepared the state $|+_X\rangle$ or whatever.
Real physicists have a way to deal with this possibility: Mixed states, and the density matrix formalism. If you try to prepare a pure state but don't do a very good job, then you get a mixed state, for example the mixed state which has a 70% chance of being $|+_Z\rangle$ and a 30% chance of being $|+_X\rangle$.
So again, as I said at the start, they have proven the obvious fact: "If a system is in the pure state $|+_Z\rangle$ then it is definitely not in some other different pure state $|+_X\rangle$ or whatever."
With such an obvious and uncontroversial premise, how do they purport to conclude anything that is not totally obvious? Let's go to the conclusions section. They conclude that the "quantum process of instantaneous wave function collapse [is different from] the (entirely nonmysterious) classical procedure of updating a probability distribution when new information is acquired." Indeed, if a spin is in the state $|+_Z\rangle$, then acquiring new information will never put it in the state $|+_X\rangle$. You have to actually do something to the system to change a spin from $|+_Z\rangle$ to $|+_X\rangle$!! For example, you could measure it, apply a magnetic field, etc.
Let's take a more interesting example, an EPR pair in the state $(|++\rangle+|--\rangle)/\sqrt{2}$. After preparing the state, it really truly is in this specific quantum state. If we carefully manipulate it while it's isolated, we can coherently change it into other states, etc. Now we separate the pair. Someone who wants to describe the first spin as completely as possible, but has no access to the second spin, would take a partial trace like usual to get a density matrix. He then gets an email that the second spin is in state +. He modifies his density matrix to the pure state +. You will notice that their example does not show that this so-called collapse violates any laws of quantum mechanics. Their disproof is specific to pure states, and would not work in this mixed state example. Therefore, they cannot conclude that "the quantum collapse must correspond to a real physical process" in the EPR case.
One more example: A spin in state $|+_Z\rangle$, and you measure it in the X-direction. The Schrodinger equation, interpreted with decoherence theory, says that the wavefunction of the universe will coherently evolve into a superposition of (macroscopic measurement of +) and (macroscopic measurement of -). In the paper, they say this in a different way: "each macroscopically different component has a direct counterpart in reality". This is just saying the same thing, but sounds more profound. I should hope that anyone who understands decoherence theory will agree that both of the macroscopic measurements are part of the universe's wavefunction, and that the universe really does have a unitarily-evolving wavefunction even if we cannot see most of it. We rarely care, however, about the wavefunction of the universe; we care only about the branch of the wavefunction that we find ourselves in. And in that branch it is quite reasonable for us to collapse our wavefunctions and to say that the other branches are "not reality" (in a more narrow sense).
-- UPDATE --
I tried to reread the paper in the most charitable way that I can. Now, I think I was a bit too harsh above. Here is what the paper proves:
CENTRAL CLAIM: Say you have a hidden-variables theory, so when you "prepare a pure state $|\psi\rangle$", you actually prepare the state $\{|\psi\rangle,A\}$, where A is the hidden variable which randomly varies each time you prepare the state. It is impossible to have $\{|\psi_0\rangle,A\}=\{|\psi_1\rangle,A'\}$, if $|\psi_0\rangle\neq|\psi_1\rangle$. In other words, the hidden-variable-ensembles of different pure states do not overlap.
They are disproving a straw-man because there is no interpretation of quantum mechanics that asserts that the hidden-variable-ensembles of different pure states must overlap with each other. Even hidden-variable theories do not assert this. There is a so-called "statistical interpretation" in the literature (advocated by L. Ballentine), which also does not assert this.
So this is a straw-man, because nobody ever argued that hidden-variable ensembles of different states ought to overlap. But, it's not a manifestly ridiculous straw-man. At least, I can't think of any much simpler way to prove that claim. (Admittedly, I do not waste my time thinking about hidden-variables theories.) I can imagine that someone who was constructing a new nonlocal-hidden-variables quantum theory might like to know that the hidden-variable-ensembles should not overlap.
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