Let us consider the statistical interpretation of quantum mechanics, as put forth by Leslie Ballentine in Rev. Mod. Phys. 42, 358, 1970.
Recently, the PBR theorem and follow-up work (e.g. this paper) have shown that "$\psi$-epsitemic theories'' face considerable conceptual difficulties.
Is Ballentine's statistical interpretation a "$\psi$-epsitemic theory" of the kind affected by the PBR theorem and its relatives?
For convvenience, here is the definition of the "Statistical Interpretation" from page 360 in ref 1:
The different interpretations of quantum theory are most sharply distinguished by their interpretations of the concept of state. Although there are many shades of interpretation..., we wish to distinguish only two:
(I) The Statistical Interpretation, according to which a pure state ... provides a description of certain statistical properties of an ensemble of similarily prepared systems, but need not provide a complete description of an individual system.
(II) Interpretations which assert that a pure state provides a complete and exhaustive description of an individual system ...
For comparison, here are the definitions of $\psi$-ontic and $\psi$-epistemic:
$\psi$-ontic means that a given reality can only be consistent with one of the quantum model's pure states.
$\psi$-epistemic means that a given reality can be consistent with two or more of the quantum model's pure states.
Here are the definitions again, in the words of ref 2:
We call a hidden variable model $\psi$-ontic if every complete physical state or ontic state [ref] in the theory is consistent with only one pure quantum state; we call it $\psi$-epistemic if there exist ontic states that are consistent with more than one pure quantum state.
Sometimes the $\psi$-epistemic option is loosely described as "the quantum state represents knowledge," or something like that, but that's not a useful definition. Of course the state represents knowledge. Even in deterministic classical physics, when we choose an initial state, we're choosing it based on our knowledge of how the physical system was prepared. And $\psi$-epistemic is not synonymous with hidden variables, either. As emphasized in ref 3, $\psi$-ontic interpretations may also involve hidden variables.
To avoid ambiguity, I'll use the definitions that were highlighted above. In particular, a $\psi$-epistemic interpretation is one in which a given reality can be consistent with two or more of the quantum model's pure states.
According to the definitions quoted above, the thing that ref 1 calls the Statistical Interpretation does not commit to either option. The Statistical Interpretation only says that the quantum model's state is an incomplete description of the physical system. It does not require that a given reality is consistent with only one pure quantum state (the $\psi$-ontic option), nor does it require that a given reality is consistent with more than one pure quantum state (the $\psi$-epistemic option). It only requires that a given pure quantum state is consistent with more than one reality.
However, the very next section in ref 1 (namely section 1.3) strongly suggests that the author was assuming a specific type of hidden-variable interpretation, despite the paper's stated goal of "providing a sound interpretation using a minimum of assumptions." Section 4.1, which is about measurement, again strongly suggests a prejudice: the last sentence in that section essentially claims that the act of measurement should merely reveal a pre-existing property of the thing being measured. That assumption is not necessary and is arguably not even viable. At the very least, it deviates from the paper's stated goal.
So we should probably distinguish between two different questions:
Is the definition of the Statistical Interpretation that ref 1 spells out in section 1.2 $\psi$-epistemic? The answer to that question is no. The definition spelled out in section 1.2 does not commit to either option, neither $\psi$-epistemic nor $\psi$-ontic.
Was the author's unacknowledged personal mental picture of the Statistical Interpretation $\psi$-epistemic? The answer to that question is I don't know, but the rest of ref 1 strongly suggests that the author had something in mind that was considerably more presumptuous than what the definition in section 1.2 actually says.
References:
Ballentine (1970), "The Statistical Interpretation of Quantum Mechanics" Rev. Mod. Phys. 42: 358
Harrigan and Spekkens, "Einstein, incompleteness, and the epistemic view of quantum states" (https://arxiv.org/abs/0706.2661)
Leifer, "Is the Quantum State Real? An Extended Review of $\psi$-ontology Theorems" (https://arxiv.org/abs/1409.1570)
