Is there a conflict, or is there not a conflict between the Pusey-Barrett-Rudolph (PBR) theorem and the information theory interpretation? In the wikipedia article, it says that the PBR theorem sort of rules out the psi epistemic interpretations. I want to know, is this the end of the information theory interpretation and relational interpretation?
I am thinking that there is no conflict. The statement of this theorem says that one physical reality is not consistent with multiple pure states. But psi epistemic models do not
attribute different pure states to the same physical situation, do they?
For example, in the Wiger's friend experiment, the information theory/relational interpretation says that the friend observes a collapsed state, say $|\text {spin up}\rangle$. But Wigner will describe the experiment using something like $|\text{spin up, friend measured up}\rangle +|\text{spin down, friend measured down}\rangle$.
So it is true that Wigner and Wigner's friend are using different states, but they're not describing the same physical situation. Wigner's friend is only describing the state of the particle. But Wigner is describing the joint system of his friend and the particle.
Is this correct? Is there a conflict or not, between the PBR theorem and the information theory/ relational interpretation?
 A: As with all such results, understanding this requires a careful consideration of the underlying mathematical framework and a knowledge of how it relates to the intuitive concepts under discussion. In particular, while it makes statements like "$\psi$-epistemic views are inconsistent with the probabilities described by QM", the term "$\psi$-epistemic" does not mean a mere equation of the wave function with "knowledge" or "information" but rather a specific mathematical model of that intuitive idea, and that mathematical model is not closed to objection. Indeed, I'll outline an objection here, that I am not sure has been raised before (but if someone has, I'd gladly accept the citation!).
PBR bascially starts from the following idea, which is a sort of extension of the setup used in Bell's Theorem. Suppose that a quantum system has associated to it some "objective" characteristics, or state, $\lambda$, drawn from a possibility set $\Lambda$. In Bell's theorem, the idea being tested is that $\lambda$ determines each measurement result - that is, for each (for simplicity) yes/no projector operator $\hat{\Pi}$ representing an elementary question to be asked about the system, $\lambda$ directly determines it answer, i.e. there is a function from the cross product of $\Lambda$ and the set of all such projectors to $\{ 0, 1 \}$, such that if we knew $\lambda$, we would know what every measurement would give in advance.
PBR softens this a bit. It doesn't presume that $\lambda$ necessary determines the measurement results by itself, but does nonetheless presume that each $\lambda$, together with a question $\hat{\Pi}$, will determine a probability for that question to be "yes", i.e. $P(\hat{\Pi}|\lambda)$, to use a notation evocative of conditional probability. It then assumes $\lambda$ is wholly set by the preparation procedure, and moreover that two different procedures set to prepare the same pure quantum state $|\psi\rangle$ prepare $\lambda$ in the same way, i.e. to each such $|\psi\rangle$, we can associate a different preparation probability measure $\mu_\psi$ on $\Lambda$, viz. the probability of getting a "yes" to some $\hat{\Pi}$ is
$$P(\text{$\hat{\Pi}$ gives "yes"}) = \int_{\lambda \in \Lambda} P(\hat{\Pi}|\lambda)\ d\mu_\psi$$
where the integral is meant in the sense of Lebesgue. Note that if the inner probability is definitive for all questions (i.e. either $0$ or $1$), then the probability to get an answer is simply that that the preparer prepared the system with that answer, i.e. the same as the setup of Bell's Theorem.
The argument then goes from there to consider a joint measurement on two independently prepared systems, and moreover shows how this measurement can be realized with real quantum systems, and shows that distinct quantum states must "partition"(*) the putative set $\Lambda$. The "$\psi$-epistemic" view is specifically defined here as the view that there could exist a pair of quantum states $|\psi_1\rangle$ and $|\psi_2\rangle$ such that the objective state $\lambda$ could belong to both, that is, knowing one or the other would be consistent with the same $\lambda$, by analogy with classical ignorance-interpreted probability distributions, where that you can assign two different probability distributions despite only 1 given state of reality being the case at a given time.
So does this rule out an informational interpretation? It depends: If you take P, B, and R's setup above as defining what you mean by "informational interpretation", then yes, it does. However, is that the only consistent informational interpretation? I would argue it does not, and will give you two reasons to think this below.
For one thing, and as I pointed out in a comment here: there are other, and arguably better-behaved, mathematical quantum formalisms in which a distinction between pure and mixed states does not have to be drawn so explicitly as in the Hilbert formalism, but instead emerges naturally from a common description. In particular, the system of $C^{*}$-algebras reveals that pure states are essentially extreme points on a convex hull of mixed quantum states. This exists in Hilbertian QM as well, if you just stick entirely to density operators and pretend vectors don't exist. Here, the vectors go away and thus we just have one set of "states". As I said before, an "informational interpretation", most generally, simply means assigning the semantic value of "agent knowledge" to the quantum state. In this case, then it follows both mixed and pure states represent knowledge. It is just that, under the PBR setup, pure states represent enough knowledge to separate at least different classes of real states of the system. There is no reason an "information" referent cannot become coincident in some sense with actual reality.
The second, more subtle, objection is the one I'm not so sure has been made before. It concerns a step that is easy to miss within the PBR argument, and that is that the move from the individual systems to the joint system assumes a passage from individual objective state spaces $\Lambda_1$ and $\Lambda_2$ to a total joint space of $\Lambda_1 \times \Lambda_2$. Besides providing another attack vector on the theorem by itself (though I believe some have followed at least direct approaches down that line and not found them very fruitful), the more interesting line of objection is what it says regarding the assumptions underlying the Bell-like framework I just described. In particular, to what extent can we consider the "state" $\lambda$, or even for that matter, $|\psi\rangle$, to truly belong solely to the system? Indeed, here's a simple thought experiment as to why we should think that frameworks of the kind P, B, and R use shouldn't be used, and it requires us to remember, as seem strangely often forgotten, that quantum mechanics, as the name suggests, is a theory of mechanics - in all this, let's not lose sight of basic physics principles! In particular, we can shift physical reference frames - for all this stuff about talking about "agents", "knowledge" and "observer effects" and what not, it seems often ignored is the question of a simple change of coordinates of the kind you learn about in your first course in Newtonian mechanics.
Suppose we have a setup located far out in the depths of space, away from gravity and other perturbing forces. There is a single electron, surrounded by some probing apparatus which, depending on how it's oriented, will measure a different component of the electron's spin angular momentum vector. We consider this apparatus to include enough capabilities it can be treated as a quantum agent, e.g. maybe it's running an AI program. The whole thing is prepared so that it regards the electron it has as being in a state like $\left|\uparrow\right\rangle$.
Now, here's something. This agent has reaction wheels attached to it. It can rotate - change its orientation - without influencing the electron. Suppose it uses them to execute a 90-degree repositioning maneuver around it. Presumably, $\lambda$ for the electron should not change, right? Yet in order to calculate the correct probabilities for the measurement, it must now assign it the Hadamard state
$$|+\rangle = \frac{\left|\uparrow\right\rangle + \left|\downarrow\right\rangle}{\sqrt{2}}$$
and will obtain $\uparrow$ or $\downarrow$ with probability $\frac{1}{2}$. Yet nothing changed about $\lambda$! This situation cannot easily be accounted for in the PBR framework! Either we have to assume that $\lambda$ is somehow strangely altered "spookily" by the rotation, or we have to assume the set of questions the agent is asking is changing. The trouble with the second option is that it violates fundamental physical symmetry laws: namely, it tells us there is a preferred orientation in space, against which those questions are defined, because no such transformation appears in the PBR framework itself - the operators all remain the same, only the quantum state $\left|\psi\right\rangle$ changes.
Hence, the thesis I would suggest is that, in fact, our quantum pure state assignment $|\psi\rangle$ actually does not just capture the "state of the system" alone, but instead reflects also some stuff about the situation of our agent viz. that system. In particular, there is no reason that $P(\hat{\Pi}|\lambda)\ d\mu_\psi(\{ \lambda \})$ should be considered sufficient to generate the probability given by QM to begin with, because it also needs conditioning upon some further aspects of the world external to the system! And that is tacitly stamped out because the movement to the Cartesian product when considering the joint systems is in effect to limit the referents solely to the systems themselves.
And indeed, Rovelli's relational interpretation is very natural for dealing with this scenario, and cannot be given an ontology in the PBR framework, because it does not assign objective states $\lambda$ to single systems alone, but instead to pairs of systems. In particular, relational ontology might look something like this: if we have three systems, $a$, $b$, and $c$ where $b$ and $c$ can act as agents observing $a$, then we would say the pairs $(b, a)$ and $(c, a)$ (where I've written it in the form $(\text{agent}, \text{system})$) have (different!) objective states $\lambda$, but not $a$ itself, nor presumably $b$ or $c$ by themselves (very interesting metaphysical implications). (Note that we could add a "self-state" to pairs like $(a, a)$, but it will not be the one observed by any external measurer.) Most particularly, in the PBR setup we have the systems $a$ and $b$, individual measurers $m_a$ and $m_b$, and joint measurer $m_j$. Then the pairs $(m_a, a)$, $(m_b, b)$, $(m_j, a \otimes b)$ could (and will!) all have different states, too. But also note that Rovelli doesn't say $\Lambda$ is distinct from the Hilbert space $H$ of any of the systems - they could, in fact, be one and the same.
Going back to our example, the mechanical rotation process I described would then be understood as changing the state of the relational pair involving the measuring agent and measured system, but changes nothing about the state of that measured system individually.
A: OKay so PBR is solely a statement about hidden variable theories, i.e. theories which say that the pure state is an incomplete description of physical reality, and that a hidden state provides the complete description.
Since the relational interpretation and the information theoretic interpretation do not assert the existence of any hidden variables, I think PBR is indeed not in any conflict with them
A: The PBR theorem essentially says that different quantum states have different observable consequences. One ontological reality cannot be correctly described by two distinct quantum states - one of them must be false in a way that can be experimentally demonstrated.
In the case of Wigner's friend, this is fairly obvious. Wigner's friend says the quantum state contains only one outcome, $\left| \uparrow \right>$ say, Wigner says it contains both. If Wigner's state is correct, then $\left| \downarrow \right>$ is still possible with non-zero probability, even after his friend has asserted that $\left| \uparrow \right>$ is the only possibility. The two quantum states make different physical predictions about what will be observed.
(Your suggestion that Wigner's friend only asserts the state of the particle and Wigner asserts the state of particle and friend doesn't work. Wigner's friend knows his own state, too. So Wigner sees the complete state as $\left| \uparrow | O_\uparrow \right>$ and Wigner sees it as $\alpha\left| \uparrow , O_\uparrow \right>+\beta\left| \downarrow , O_\downarrow \right>$. The friend can choose to leave the observer part out, but it is still a part of his full knowledge of the state of the world.)
Epistemic interpretations say that these differences merely reflect the different state of knowledge of each observer. An observer might have only partial information, the quantum state they assert on the basis of what they know may be different from the true ontological state. So if Wigner's friend has already collapsed the wavefunction, then Wigner is simply wrong in thinking both options are still open.
There is a distinction between versions that suggest there is no 'true' ontology behind the scenes, only the epistemic information from observations - like a game of 20 questions where the world answers 'yes' or 'no' at random, consistent with previous answers - and interpretations where there is an ontological truth behind it all but it is something forever inaccessible to observers.
Since the PBR theorem assumes there is a single ontological state of the world, I don't think the former are directly invalidated. PBR contradicts non-realist interpretations, but only by prior assumption. It depends on whether you think PBR's assumption is reasonable. Tastes may differ!
And since an epistemic interpretation allows that an epistemic estimate based on partial information may be wrong about the true state, and the probabilities that arise, adherents are not too distressed to discover that this is so.
The relational interpretations, on the other hand, get round the problem by asserting that ontological reality is observer-dependent. Quantum states describe the relation between observer and observed. A system does not have to show the same face to all observers. So again, the assumption of a single ontological reality and the need for a single set of consistent predictions is challenged.
But I think there is a simpler viewpoint, which is to recognise that quantum mechanical interpretations are by their nature not subject to confirmation or refutation. The unitary evolution of quantum mechanics predicts that interaction between systems will lead to a superposition of mutually-invisible alternative outcomes. Since these other outcomes are unobservable even in principle, we can have no observational evidence that they are there, (or not there). They might all continue to exist, unobserved and unobservable (many worlds), or they might all instantly disappear (Copenhagen), or some of them might partially disappear for some observers and not others (relational), or anything else (explode into showers of unicorns and party balloons...). So long as you make all the same predictions about the bits you can see, there is no way to experimentally distinguish the hypotheses about what you can never see.
The only grounds on which you can pick or dismiss one are aesthetic. Does it have nice properties, like being realist, deterministic, local, causal, parsimonious, simple, complete, materialist, non-mystical, etc.? Does it explain things that other hypotheses simply have to assert as axioms? Does it explain when, how, and why the other alternatives disappear? Is it easy to imagine? Easy to calculate with? Easy to explain?
So really, something like the PBR theorem can never refute any particular interpretation (except by refuting quantum mechanics itself), but you can reasonably ask whether, given the consequences, the assumptions it makes and the reasons for them have more or less aesthetic merit than those of particular interpretations.
A: 
psi epistemic models do not attribute different pure states to the same physical situation, do they?

For better or worse, it seems to be standard to use "$ψ$-ontic" to describe models where the quantum state vector is a function of the state of the world, and "$ψ$-epistemic" to describe models where it is not. So $ψ$-epistemic models do have physical states that are shared by more than one quantum state, by definition.
It's a little hard to say whether a given model is $ψ$-epistemic or not, because we don't know the correct $ψ$. If a system starts in the state $|{\rightarrow}\rangle|\text{experimenter}\rangle$, and according to the many-worlds interpretation evolves from there to $$\frac{|{\uparrow}\rangle|\text{experimenter saw }{\uparrow}\rangle + |{\downarrow}\rangle|\text{experimenter saw }{\downarrow}\rangle} {\sqrt2}$$
but according to an explicit-collapse model evolves to either the state $|{\uparrow}\rangle|\text{experimenter saw }{\uparrow}\rangle$ or the state $|{\downarrow}\rangle|\text{experimenter saw }{\downarrow}\rangle$, then presumably at most one of those models can be $ψ$-ontic, but we don't know which one.
The argument in the PBR paper nominally applies to any model that is $ψ$-epistemic, but it assumes that the measurements needed to obtain a contradiction can actually be carried out. Measurements involving superpositions of experimenter states can't be done, so the argument doesn't really rule out anything involving states of experimenters. Even if treated as a purely theoretical argument about consistency with QM, it still doesn't, because the many-worlds and explicit-collapse models mentioned above are both considered to be "consistent with QM" by many people.
The relational interpretation goes much farther in denying that there even is a correct $ψ$ in these situations, so I'm not sure it can even be fit into the $ψ$-ontic/epistemic classification, but in any case there's no way it can be said to be ruled out by PBR's argument.
