The idea is:

Suppose all wavefunctions encode some constant, sub-unity fraction of the information about the 'true state' - that all wavefunctions are equally specific. Then we should be able to tease out more of that information with some very basic experiments (edited to clarify: by measuring in multiple ways - not that the new state would be intrinsically better constrained). This contradicts the results of the many many instances of these experiments.

So you can escape by supposing that it's possible to prepare a wavefunction that has more or less information about the 'true state'.

It's nice to have yet another very strong constraint to put on global hidden variables, but nothing could ever be total proof.

~~~~ edited to add: But if you pull that escape route, then it's trivial to perform (by which I mean, they have already been done zillions of times) additional experiments showing that quantum states with different distributions of 'real' states (as we just allowed by the escape route) actually have identical dynamics.

The more I think about this, the less wiggle room I see.

Also note the companion article "The quantum state can be interpreted statistically," by two of the same authors. A previous title for this paper was "The quantum state cannot be interpreted statistically," by the way.

This heavily implies that what's being argued here is deeply philosophical, hinging crucially on the interpretation of "statistically" and therefore has a high probability of being incorrect.

I'm withholding my judgment, for the moment.

So, it turns out it is more than "just" a useful mathematical tool after all. Does this confirm the universe basically runs on maths?

Don't lend too much credence to a single paper and don't make far-reaching conclusions just yet. This is a nice result, but it is not of the same level as, say, the Bell inequalities.

From the article:

The argument depends on few assumptions. One is that a system has a “real physical state” – not necessarily completely described by quantum theory, but objective and independent of the observer. This assumption only needs to hold for systems that are isolated, and not entangled with other systems. Nonetheless, this assumption, or some part of it, would be denied by instrumentalist approaches to quantum theory, wherein the quantum state is merely a calculational tool for making predictions concerning macroscopic measurement outcomes.

Skimming the article I haven't found a precise definition of what having a real physical state means (which would be interesting and important in such a discussion, especially given the authors scare quotes around the term). Nevertheless, the assumption that the system has a state is quite a strong one. If one wants to deny that the wave function is real (whatever it means) one can easily deny the reality of states, and treat QM only as a tool predicting which sequences of measurements can happen and how probably.

I have an (undergraduate) degree in physics, or will at the end of the semester, and either I don't fully understand the paper after a second read-through, or it is wrong. The claimed contradiction doesn't seem to be particularly damning: if the measured state is excluded from being in a |0>|0> initial state, that doesn't mean that the two cannot have started in a different state, and the excluded state will change in later measurements anyway, so the measurement is not telling you anything new.

(Motl's explanation is harsh and caustic, but here I agree with him in interpreting the paper.)

Edit: I agree with Motl's explanation of why the exclusion of 1 of the 4 combined states is not problematic.

http://motls.blogspot.co.uk/2011/11/nature-hypes-anti-qm-crackpot-paper-by.html