# Significance of Pusey-Barrett-Rudolph (PBR) Theorem

I recently read the PBR Theorem and from what I understand it addresses one of the old problems regarding quantum state completeness, one for which there are three possibilities (correct me if am wrong here):

• A physical state $$\lambda$$ corresponds to several quantum states $$\Psi_{i}$$

• Multiple physical states $$\lambda_{i}$$ can correspond to a single quantum state $$\Psi$$

• There are no physical states, implying that the quantum state fully encodes a system

The PBR seems to rule out the first possibility, implying that if you allow for physical states to exist, then there is a unique quantum state faithfully representing that system.

We then know that the second possibility can't be right, and the quantum state is not a statistical description of a system in any realist model of QM (that is, one allowing for physical states to exist).

Now, suppose we choose the third option, my question is which system is the quantum state fully encoding if there are no physical states? How do we prove that this encoding is unique? It seems to me that this question is well-posed and provable only in a $$\lambda$$ model, like the PBR shows, and not otherwise.

• "We then know that the second possibility can't be right" Can you further elaborate why (I thought the paper only eliminates the first)? Wouldn't Bohmian mechanics be an example of the second possibility (more than one possible particle configuration is compatible with a given wavefunction)? Commented Jan 2 at 13:25
• If I am not mistaken, two different wave functions in Bohm's theory would produce two different and unique distributions of position variables, because the extra dynamical equation is of first order. Commented Jan 2 at 13:35

The first option is called psi-epistemic, and is - as you say - ruled out by PBR. The second you have is slightly off. The second is actually called psi-ontic and says there is only one quantum state associated with a one particular physical state. However, it may or may not be complete, and is usually taken to be incomplete. It is also called psi-supplemented. The third option is to be also psi-ontic, but is considered complete and is often referred to as psi-complete. The following pre-PBR reference explains this in more detail.

https://arxiv.org/abs/0706.2661v1

Einstein, incompleteness, and the epistemic view of quantum states

“We explore a distinction among hidden variable mod- els of quantum theory that has hitherto not been suffi- ciently emphasized, namely, whether the quantum state is considered to be ontic or epistemic. We call a hid- den variable model ψ-ontic if every complete physical state or ontic state [1] in the theory is consistent with only one pure quantum state; we call it ψ-epistemic if there exist ontic states that are consistent with more than one pure quantum state. In ψ-ontic models, dis- tinct quantum states correspond to disjoint probability distributions over the space of ontic states, whereas in ψ-epistemic models, there exist distinct quantum states that correspond to overlapping probability distributions. Only in the latter case can the quantum state be consid- ered to be truly epistemic, that is, a representation of an observer’s knowledge of reality rather than reality itself.”

Some of the more popular, modern interpretations of QM assert that the quantum state is representative of our knowledge, and that experiments update that knowledge. There are a variety of names for some of these interpretations, and some of them have different ways of representing that idea. Some of those include QBism and Ensemble/Statistical interpretations.

PBR attempts to counter that view, by demonstrating - through some very reasonable assumptions - some basic contradictions with the epistemic view.

I personally think that the PBR theorem is very strong, as a no-go theorem. It goes along nicely with standard QM, especially stripped down versions in which the underlying mechanisms are not detailed.

On the other hand: it is a bit difficult to precisely assign interpretations to one group or the other. The reason for that being that proponents of each interpretation usually deny any opposing proofs, often with a lot of unconvincing hand waving.

So to finally answer your question about the 3rd option, which is psi-complete: we need to demonstrate there are no hidden or supplemental variables which explain the outcomes of individual experimental trials.

• Thanks for the answer, what I am still confused about is how you can explain the outcomes of individual experiments in a model that doesn't have a notion of physical system (the third option) Commented Jan 3 at 11:36