The quantum state cannot be interpreted statistically?

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.

I thought this was an interesting quote by Matt Leifer on his most recent comment on his blog:

What does Leifer mean by "nonlocality"?

At the least, some type of superluminal "influence", I think.

That's understandably and, imho, unacceptably, vague. Nonlocality is defined by some (most? ... I don't know) quantum physicists as referring to entangled quantum states, which might ultimately refer to ftl propagations or not.

I think it’s a deep dilemma, and the resolution of it will not be trivial; it will require a substantial change in the way we look at things. But I would say that the cheapest resolution is something like going back to relativity as it was before Einstein, when people like Lorentz and Poincare thought that there was an aether -a preferred frame of reference-but that our measuring instruments were distorted by motion in such a way that we could not detect motion through the aether...that is certainly the cheapest solution. Behind the apparent Lorentz invariance of the phenomena, there is a deeper level which is not Lorentz invariant...what is not sufficiently emphasized in textbooks, in my opinion, is that the pre-Einstein position of Lorentz and Poincar´e, Larmor and Fitzgerald was perfectly coherent, and is not inconsistent with relativity theory. The idea that there is an aether, and these Fitzgerald contractions and Larmor dilations occur, and that as a result the instruments do not detect motion through the aether - that is a perfectly coherent point of view...The reason I want to go back to the idea of an aether here is because in these EPR experiments there is the suggestion that behind the scenes something is going faster than light. Now if all Lorentz frames are equivalent, that also means that things can go backwards in time...[this] introduces great problems, paradoxes of causality, and so on. And so it is precisely to avoid these that I want to say there is a real causal sequence which is defined in the aether.”

Valentini’s Aristotelian spacetime: Galilean invariance not a fundamental symmetry of the standard low-energy pilot-wave theory. The search for a Lorentz-invariant extension thus seems misguided. In Valentini’s view, the difficulties encountered in such a search are no reflection on the plausibility of the pilot-wave theory. Rather, they show that the theory is not being interpreted correctly. Pilot-wave theory then has a remarkable internal logic - both structure of dynamics, and operational possibility of nonlocal signalling away from equilibrium (see later) independently point to existence of natural preferred state of rest.

I'm not sure what you are asking.

If one buys Leifer's argument, it's pretty clear which models PBR scraps.

With respect to non-locality, some physicists (e.g. Bell, Maudlin, Laudisa, Norsen, etc.) interpreted Bell's theorem as already implying non-locality (ftl) irrespective of "realism" issues.

Others, however, did not interpret Bell's theorem in that way.

I think it has to be vague (e.g. "influence") because some have argued that ftl does not imply incompatibility with relativity since it may depend on which interpretation of relativity is true.

A Lorentzian interpretation of relativity (single preferred frame) is compatible with non-locality.

Does this mean just some finite v>c or instantaneous influence? I think it implies the latter.

Well, no, I don't buy Leifer's take on things.

Why?

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.

In short, they try to show that there is no lambda satisfying certain properties. The problem is that the CRUCIAL property they assume is not even stated as being one of the properties, probably because they thought that property was "obvious". And that "obvious" property is today known as non-contextuality. Indeed, today it is well known that QM is NOT non-contextual. But long time ago, it was not known. A long time ago von Neumann has found a "proof" that hidden variables (i.e., lambda) were impossible, but later it was realized that he tacitly assumed non-contextuality, so today it is known that his theorem only shows that non-contextual hidden variables are impossible. It seems that essentially the same mistake made long time ago by von Neumann is now repeated by those guys here.

Let me explain what makes me arrive to that conclusion. They first talk about ONE system and try to prove that there is no adequate lambda for such a system. But to prove that, they actually consider the case of TWO such systems. Initially this is not a problem because initially the two systems are independent (see Fig. 1). But at the measurement, the two systems are brought together (Fig. 1), so the assumption of independence is no longer justified. Indeed, the states in Eq. (1) are ENTANGLED states, which correspond to not-independent systems. Even though the systems were independent before the measurement, they became dependent in a measurement. The properties of the system change by measurement, which, by definition, is contextuality. And yet, the authors seem to tacitly (but erroneously) assume that the two systems should remain independent even at the measurement. In a contextual theory, the lambda at the measurement is NOT merely the collection of lambda_1 and lambda_2 before the measurement, which the authors don't seem to realize.

I had a brief exchange of e-mails with the authors of that paper. After that, now I am even more convinced that I am right and they are wrong. Here are some crucial parts of that exchange, so that you can draw a conclusion by yourself:

> Prof. Barrett:
> Briefly, the vectors in Eq.(1) are entangled, yes but they don't represent
> the state of the system. They are the Hilbert space vectors which
> correspond to the four possible outcomes of the measurement.

Me (H.N.): But in my view, the actual outcome of the measurement (i.e., one of those in Eq. (1) ) DOES represent the state of the system. Not the state before the measurement, but the state immediately after the measurement. At the measurement the wave function "collapses", either through a true von Neumann collapse, or through an effective collapse as in the many-world interpretation or Bohmian interpretation.
...

> Prof. Barrett:
> The assumption is that the probabilities for the different outcomes of
> this procedure depend only on the physical properties of the systems at a
> time just before the procedure begins (along with the physical properties
> of the measuring device).

Me (H.N.): Yes, I fully understand that if you take that assumption, you get the conclusion you get. (In fact, that conclusion is not even entirely new. For example, the Kochen-Specker theorem proves something very similar.) But it is precisely that assumption that I don't find justified. Any measurement involves an interaction, and any measurement takes some time (during which decoherence occurs), so I don't think it is justified to assume that the measurement does not affect the probabilities for the different outcomes.

In short, to make their results meaningfull, a correct title of their paper should be changed to "The quantum state cannot be interpreted non-contextually statistically" But that is definitely not new!

The issue of measurements causing a disturbance is not relevant here since we are only considering a simple prepare-and-measure experiment. If we were concerned with what happens after the measurement then it would be relevant, but this is not involved in the PBR scenario.

There is no assumption in the PBR paper that xi^k_p depends only on the projector. It may also depend on the other projectors in the measurement, i.e. it may be different for different measurements that share a common projector. However, the proof of the PBR theorem only makes use of a single measurement, so it doesn't get into trouble with the KS theorem in any case.

Since http://arxiv.org/abs/1201.6554 came out, we now know that psi-ontology and contextuality are definitely separate issues, since a psi-epistemic theory can be obtained for any Hilbert-space dimension, whereas a noncontextual theory cannot. This also shows that the factorization assumption is crucial in the PBR proof.

We have seen that the PBR result can be used to establish some known constraints on hidden variable theories in a very straightforward way. There is more to this story that I can possibly fit into this article, and I suspect that every major no-go result for hidden variable theories may fall under the rubric of PBR. Thus, even if you don’t care a fig about fancy distinctions between ontic and epistemic states, it is still worth devoting a few braincells to the PBR result. I predict that it will become viewed as the basic result about hidden variable theories, and that we will end up teaching it to our students even before such stalwarts as Bell’s theorem and Kochen-Specker.

... some have argued that non-locality does not imply incompatibility with relativity since it may depend on which interpretation of relativity is true.

I think it’s a deep dilemma, and the resolution of it will not be trivial; it will require a substantial change in the way we look at things. But I would say that the cheapest resolution is something like going back to relativity as it was before Einstein, when people like Lorentz and Poincare thought that there was an aether -a preferred frame of reference-but that our measuring instruments were distorted by motion in such a way that we could not detect motion through the aether...that is certainly the cheapest solution. Behind the apparent Lorentz invariance of the phenomena, there is a deeper level which is not Lorentz invariant...what is not sufficiently emphasized in textbooks, in my opinion, is that the pre-Einstein position of Lorentz and Poincar´e, Larmor and Fitzgerald was perfectly coherent, and is not inconsistent with relativity theory. The idea that there is an aether, and these Fitzgerald contractions and Larmor dilations occur, and that as a result the instruments do not detect motion through the aether - that is a perfectly coherent point of view...The reason I want to go back to the idea of an aether here is because in these EPR experiments there is the suggestion that behind the scenes something is going faster than light. Now if all Lorentz frames are equivalent, that also means that things can go backwards in time...[this] introduces great problems, paradoxes of causality, and so on. And so it is precisely to avoid these that I want to say there is a real causal sequence which is defined in the aether.”