I haven't checked the calculations, but I like the logic very much. I'm kicking myself for not having tried harder to precisely formalize what the authors refer to as the "statistical interpretation" (note: this is quite a confusing terminology for most people -- see Further comments below) of the quantum state. Apparently, once you formalize this interpretation, it is easy to prove that it has to disagree with the predictions of ordinary quantum theory.
This "statistical interpretation" (e.g., that the wavefunction, or quantum formalism, only describes the knowledge state of the observer and does not correspond to physical reality) is the last shaky dodge of those who are against the reality (or correspondence to reality) of the wavefunction. The latter has always seemed to me the natural first interpretation of the formalism, subject, of course, to further analysis.
The quantum state cannot be interpreted statistically
Matthew F. Pusey, Jonathan Barrett, Terry Rudolph
Quantum states are the key mathematical objects in quantum theory. It is therefore surprising that physicists have been unable to agree on what a quantum state represents. There are at least two opposing schools of thought, each almost as old as quantum theory itself. One is that a pure state is a physical property of system, much like position and momentum in classical mechanics. Another is that even a pure state has only a statistical significance, akin to a probability distribution in statistical mechanics. Here we show that, given only very mild assumptions, the statistical interpretation of the quantum state is inconsistent with the predictions of quantum theory. This result holds even in the presence of small amounts of experimental noise, and is therefore amenable to experimental test using present or near-future technology. If the predictions of quantum theory are confirmed, such a test would show that distinct quantum states must correspond to physically distinct states of reality.
Here's what Nature News had to say:
Quantum theorem shakes foundations: The wavefunction is a real physical object after all, say researchers.
There seems to be widespread misunderstanding of what the authors are trying to do in this paper.
They are not trying to refute qm or the standard rules of calculation (e.g., Born rule). Perhaps their use of the term "statistical interpretation" is unfortunate because some people seem to have jumped to the conclusion that they claim to prove qm is deterministic or non-probabilistic. That is not the case.
They are addressing a particular interpretation of qm. This interpretation says: there is an underlying physical reality, but the state Psi only describes an observer's knowledge about that underlying reality. Psi is not itself a direct representation of that reality. ("Psi is not real".) I would classify this as a variant of Copenhagen; its proponents sometimes refer to it as a "Bayesian" or "Epistemic" interpretation. I prefer to call it the "Mysterian" interpretation: reality is some vast mysterious thing (never specified!), Psi only characterizes the observer's mental state; collapse of the wavefunction is simply a Bayesian update of the mental state.
Mysterian/Bayesian: "The reduction of the wavefunction takes place in the consciousness of the observer ... because the state is a construct of the observer's mind and not an objective property of the physical system."
Many Worlder: "The wavefunction is real (i.e., a direct representation of physical reality), but it does not collapse."
Note, both groups try to avoid the possibility that Psi is real and collapses. But see Weinberg's recent preprint for an attempt to understand that possibility: http://arxiv.org/abs/1109.6462
A modern proponent of the Mysterian point of view is Chris Fuchs. I would be very interested to hear his reaction to this paper. But Rob Spekkens (quoted in the Nature article) also thinks along these lines, and he seems to believe that the (lambda, q) formalization of Mysterianism captures something useful. I am still pondering it myself.
Technically, the (lambda, q) formalization describes a model in which (i) there is an underlying reality (some Mysterians apparently do not actually believe this) and (ii) the state vector Psi does not describe the underlying reality but rather an observer's knowledge about it.
The fact that a given underlying reality lambda has probability q of being consistent with two different preparations of a state, which each yield different pure states phi_0 and phi_1 (their notation), is meant to capture (i) and (ii) above. Remember that to a Mysterian the pure state is a description of a state of knowledge, not of reality. So nonzero q means that two different states of knowledge (preparations) are consistent with the same underlying state of reality.
These Fuchs slides might be of use in understanding the mysterious Mysterians: Being Bayesian in a Quantum World (I am a Bayesian, who lives in a quantum world, but not a Mysterian :-)
This blog post by Matt Leifer is very clear and gives the context for the paper in the qm foundations subfield.