However I yet have some difficulties with the PBR result, have posted them here: http://goo.gl/EjCgi. Pleas take a look and help me in resolving.

“Many quantum physicists have suggested that a quantum state does not represent reality directly, but rather the information available to some agent or experimenter. This view is attractive because if a quantum state represents only information, then the collapse of the quantum state on measurement is possibly no more mysterious than the Bayesian procedure of updating a probability distribution on the acquisition of new data. In order to explore the idea in a rigorous setting, we consider models for quantum systems with probabilities for measurement outcomes determined by some underlying physical state of the system, where the underlying state is not necessarily described by quantum theory. A quantum state corresponds to a probability distribution over the underlying physical states, in such a way that the Born rule is recovered. We show that models can be constructed such that more than one quantum state is consistent with a single underlying physical state-in other words the probability distributions corresponding to distinct quantum states overlap. A recent no-go theorem states that such models are impossible. The results of this paper do not contradict that theorem, since the models violate one of its assumptions: they do not have the property that product quantum states are associated with independent underlying physical states.”

The quantum state can be interpreted statistically
http://lanl.arxiv.org/pdf/1201.6554.pdf

Well, let me back up a bit. It is a consequence of PBR under one additional assumption, which is that hidden variable theories are convex under mixing, i.e. if quantum states \(\Ket{\psi_j}\) are represented in a HVT by the distributions \(\mu_j(\lambda)\), and if I prepare the state \(\rho = \sum_j p_j \Ket{\psi_j}\Bra{\psi_j}\) by generating a classical random variable (e.g. by flipping coins, rolling dice, etc.) with outcome probabilities \(p_j\) and then preparing \(\Ket{\psi_j}\) when the outcome is \(j\), then this results in the hidden variable distribution \(\mu(\lambda) = \sum_j p_j \mu_j(\lambda)\). Formally, one could imagine theories in which this wasn’t true, e.g. the coin flipping mechanism is somehow correlated with the ontic state of the quantum system, but it would be pretty loopy to do so.

With this convexity assumption, the fact that two ensembles represented by the same density matrix must be represented by different distributions in a hidden variable theory was already known prior to the PBR theorem, and was called “preparation contextuality” by Rob Spekkens (see http://arxiv.org/abs/quant-ph/0406166). The PBR theorem implies an even stronger result, which is that the probability distributions representing any two ensembles with no pure states in common (as in the PUP vs. CUP example) can have no overlap.

Personally, although I am confident that quantum theory is not the final theory, I am skeptical that nature will choose to violate it a way that immediately solves the measurement problem. Instead, I imagine that the next theory will look even weirder than quantum theory. Therefore, attempts at modifying quantum theory that are motivated just by solving the measurement problem and nothing else are not that appealing to me. I could be wrong of course, and these approaches do deserve to be investigated and tested by the physics community, but my prior probability for them being correct is sufficiently low that I won’t be working on them myself. The exception is nonequilibruim Bohmian mechanics, as this has structure that I find interesting and is at least a very concrete proposal for post-quantum physics. However, I view this more as a conceptual investigation into the theoryspace around quantum theory rather than as a plausible candidate for future physics.

Meanwhile, you might find this interesting, I think this is a candidate for psi-epistemic realist and deterministic interpretation that the PRB theorem does not affect:
http://de.arxiv.org/abs/1112.1811

- The use of the word “statistical” is, at the end of the day, just a matter of terminology and we should be able to get past it. Once we are clear on how the authors are using it, it doesn’t really make a great deal of difference whether or not you like their usage. However, I still maintain that it was a mistake to use this terminology because it has turned a lot of people off the result on the basis of the title alone. The problem is that the “statistical interpretation of the wavefunction” is interpreted by many people as nothing more than the correctness of the Born rule. This is not an unreasonable reading of the terminology, since Max Born’s Nobel prize citation reads, “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction”. As a result, since I wrote this blog post, my email inbox has been full of people complaining that PBR do not actually disprove the Born rule and so the result must be uninteresting. I agree with the first part, but I think that failure to appreciate the significance of what it is that they actually do prove is a mistake, and the title is helping a lot of people to make that mistake.

- Regarding the use of the classical probability calculus, I am with you, I think, in believing that quantum theory is best understood as a generalization of classical probability theory. I am also with you in thinking that the foundations of probability are a mess, or rather that your average foundationally inclined physicist has a rather poor understanding of them. However, I don’t agree with your assessment of the subjective Bayesian approach. For me, it is the best possible response to the fact that we can’t solve the problem of induction by deductive means, or rather we can’t solve the statistical variant of the problem where the aim is to deduce probabilities from relative frequencies. The subjective Bayesian responds to this by falling back on a description of what an agent expects to happen on the basis of data, rather than attempting to describe what will happen. In doing so, he makes it clear that the usual description of statistical inference depends on many assumptions, e.g. exchangeability, and that no one is forced to agree that these assumptions must hold in any given situation. However, if we do have a common set of assumptions, e.g. we all adopt exchangeability, then the theory explains how observing data will end up causing us to reach agreement. Admittedly, it may be preferable to have a theory of why we reach agreement that also implies that the agreement that we reach is objectively true, as the frequentist and propensity accounts purport to maintain. However, I think that the problem of induction implies that such a theory of truth from statistics is logically impossible. Therefore, I am inclined to accept the subjective Bayesian account as the best that one can hope for. At the very least, it forces us to state the assumptions on which probabilistic assessments depend explicitly, rather than sweeping them under the carpet and pretending that they don’t exist as is usually done in objectivist accounts.

Given that I have adopted the subjective Bayesian approach, it is then necessary to determine what it means to have a generalization of probability theory, since that is how I want to understand quantum theory. It is not enough to just point to the Kolmogorov axioms and generalize them formally, as is usually done, since those axioms are not what gives probability its real-world meaning. For a subjective Bayesian, generalizing probability theory can only mean that one or more of the assumptions usually made in the decision theoretic arguments for classical probability do not hold in general. This is not a crazy thing to think, since those assumptions have been questioned already for reasons that have nothing to do with quantum theory, and a variety of generalizations have been proposed (e.g. upper and lower probabilities, or partially ordered comparative probabilities in general). Admittedly, none of these generalizations have seen widespread application, but I view it as a virtue of the subjective Bayesian approach that it leads to them. (Generally, if you are doing work on the foundations of some theory and are adopting the assumption that the theory in question must always hold at the outset then you are doing it wrong.) In the case of quantum theory, it is very easy to see which assumptions do not hold. For the Dutch book argument, it is the idea that all bets that you might make about the properties of some physical system can simultaneously be resolved. For the Savage argument, it is the idea that there is a pre-existing “state of the world” that determines the consequences of all possible actions uniquely. Depending on how one chooses to interpret quantum theory, these assumptions may not hold, e.g. they would not hold in Copenhagenish views with the notion of complimenatarity, in the many-worlds view, or in spontaneous collapse views (they do hold in Bohmian mechanics though). If we drop these assumptions then we end up with a more general theory (technically a theory of probability measures on semi-classical state spaces — this is the Greaves-Myrvold theory that I mentioned in my last comment). Both classical and quantum probability can be embedded in this structure, although we are still a long way from pinning down a C*-algebraic structure uniquely.

Given this account of how the foundations of probability need to be generalized, we can then ask whether the adoption of classical probability theory is an extra assumption that we need to make explicit in no-go theorems like Bell, PBR, etc. or whether it is really just an implication of realism. Certainly, if we are dealing with a (single universe) deterministic theory then it is not an extra assumption because this implies the “all bets can be resolved” and “state of the world” assumptions of Dutch Book and Savage. If the theory has some genuine stochasticity then things are less clear. Savage’s axiom implies determinism, and dropping it leads to Greaves-Myrvold, so it seems like we are in good shape here. However, the idea that there is some ontic degree of freedom that causes outcomes probabilistically may be strong enough to get us back to classical probability theory. I do not know for sure, so it is worth thinking about. On the other hand, Dutch book makes no assumption of determinism, since simultaneous resolvability of all combinations of bets does not imply it. One could argue it either way, but I would be inclined to say that the idea that this can be done in principle, if not in practice, is part of the definition of what we mean by “realism”, i.e. the idea that those bet outcomes are caused by something. This would imply that the use of classical probability is not an extra assumption. However, I admit that you might not find this reasoning compelling, especially if you are not a subjectivist.

- Finally, regarding the relative merits of Bell, KS, free-will theorem etc., that is certainly a topic for debate. Describing all my views on this would require a blog post of its own, so I’ll just make a couple of comments. Firstly, the KS theorem assumes outcome determinism, so one is always free to drop that assumption rather than noncontextuality as a response. In this sense, Bell, PBR, etc. are stronger because they do not assume determinism. In my opinion, Rob Spekkens has the best definition of what noncontextuality means in the absence of determinism, but his definition does involve probabilities so the perceived advantage of KS disappears. However, this doesn’t apply to the free-will theorem because restricting attention to deterministic theories can be justified by locality and convexity, as is done in Bell’s theorem. Nevertheless, the convexity argument shows that probabilities are hiding in the background of this theorem as well. As a side note, “experimental tests” of KS (with scare quotes to indicate that I am skeptical of their relevance) generally do make probabilistic assumptions, as they are based on inequalities derived in a similar way to Bell’s theorem.

Generally speaking, I think that PBR will turn out to be the strongest of the no-go results, which is why I am so keen on promoting it. I think it may imply all of the others in some suitable sense. For example, given PBR, the EPR argument is enough to establish nonlocality, without having to bother with Bell inequalities. It also pretty simply implies Rob Spekkens notion of “preparation contextuality”, but I have to admit that I haven’t been able to figure out how it is connected to the traditional KS version of contextuality as of yet.

You write:
‘ First up, I would like to say that I find the use of the word “Statistically” in the title to be a rather unfortunate choice.’
I believe they made exactly the right choice, though, because their terminology exposes the fact that they make use of standard statistical (or one should say probabilistic) reasoning in passing from the hidden states to the wave function. As such, they rule out option 1) precisely as you state it, that is,
‘Wavefunctions are epistemic and there is some underlying ontic state. Quantum mechanics is the statistical theory of these ontic states in analogy with Liouville mechanics.’
BUT: they do not rule out the first sentence by itself. The point is that, in what you quite rightly call the ‘standard Bell framework for ontological models.’ one implicitly assumes the classical probability calculus. The use of this calculus should, in my opinion, always be explicitly added to the assumptions leading to Bell’s Theorem and related results, and hence also to PBR. Your conclusion that
‘The PBR theorem rules out psi-epistemic models within the standard Bell framework for ontological models’ is correct as stated (and I applaud your precision at this point), but one cannot say (as some surely would be inclined to do) that ‘The PBR theorem rules out psi-epistemic models’ full stop.

To restate my point, your replacement of ‘statistical’ by ‘ontic’ hides an important ingredient in practically all discussions of hidden variable models underlying QM, namely the assumption of classical probability theory. Why do I find this an important point? Surely, most readers of this blog will be aware of the shaky philosophical and conceptual status of classical probability: the frequency interpretation (which would justify the axioms and hence the calculus of classical probability if it were correct) has been discredited by almost every commentator, Popper’s propensity interpretation is now widely (and rightly) seen as empty, and finally the Dutch Book arguments of the De Finetti and his followers are so human-oriented that even Bayesians and other subjectivists should raise their eyebrows in falling back on it. In any case, I would be baffled if anyone would dare to justify the statistical averaging over invisible hidden variables underlying quantum theory by a theory of betting!

To close this comment: although a detailed comparison between the strength of Bell-type theorems and Conway-Kochen-Specker type theorems has not, to my knowledge, been made in the literature, I would maintain that the latter are a priori stronger since they rely on weaker assumptions. For, as you know, the K-S and Free Will Theorems do not rely on classical or any other kind of probability at all.

Best wishes, Klaas