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?
Best Answer
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.