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The Pusey-Barrett-Rudolph theorem excludes overlapping preparation distributions for distinct pure states in an ontological model of a quantum system satisfying preparation independence and the specified quantum predictions. It excludes neither all hidden variables nor every informational use of a quantum state. The precise overlap criterion is essential: a psi-epistemic model lets at least two distinct pure-state preparations assign positive common probability mass to the same underlying states; a psi-ontic model has mutually singular measures for each distinct pair.^[ref]

Let preparation $x$ sample $\lambda \in \Lambda$ from ${\mu }_x$. A fixed quantum measurement $M$ has nonnegative response functions ${\xi }_M(j|\lambda )$ with ${\sum }_j{\xi }_M(j|\lambda )=1$. Averaging over the preparation must reproduce the Born rule. The response is fixed by $M$ and the complete underlying state, not by additional knowledge of the preparation label. No outcome determinism or Bell local causality is assumed; the joint measurement response need not factor into local responses. For independently operated preparations, preparation independence further identifies the joint state space with the appropriate product and assignsOperationally independent devices do not logically entail this ontic factorization; it is an additional physical premise.^[ref]

The two-qubit example can be checked directly. Set $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$. In computational basis order $00,01,10,11$, take the following normalized columns as an orthonormal basis:Their pairwise inner products vanish. Direct multiplication shows that the respective preparations $00,0+,+0,++$ have zero probability amplitude for respective outcomes $1,2,3,4$. This is the PBR exclusion measurement for zero and plus: the measurement does not perfectly distinguish the preparations, but each outcome rules one out.

Suppose a common submeasure $\nu$ of ${\mu }_0$ and ${\mu }_{+}$ has mass $\omega >0$; with densities one can take $\nu (\lambda )=\min \{{\mu }_0(\lambda ),{\mu }_{+}(\lambda )\}$. Preparation independence makes $\nu \otimes \nu$ a common submeasure of all four product preparations, of mass ${\omega }^2$. The zero probability for outcome $j$ under its excluded preparation and nonnegativity forceSumming and using normalization yields ${\omega }^2=0$, a contradiction. This measure formulation avoids treating a set with merely nonempty topological support as a positive-probability overlap.

For arbitrary distinct nonorthogonal pure states, write $|{\psi }_{0,1}⟩=\cos (\theta /2)|0⟩\pm \sin (\theta /2)|1⟩$, with overlap $\cos \theta$ and $0<\theta <\pi /2$. The general proof constructs an exclusion measurement on $n$ independently prepared copies when $2^{1/n}-1\le \tan (\theta /2)$. A finite $n$ always exists. The same common-submeasure contradiction then excludes positive overlap. Orthogonal preparations are already disjoint if their perfectly distinguishing measurement is reproduced. The original paper also bounds residual overlap for imperfect exclusion data; exact disjointness concerns the ideal theorem, not a finite experiment's proof of exact zero.^[ref]

The scope is consequently sharp. Bohmian mechanics is not excluded: its underlying state includes the wave function, making distinct pure preparations disjoint even though extra configuration variables remain. A psi-ontic model need not be complete with the wave function alone, and the theorem does not identify the wave function with a three-dimensional material field. Models abandoning preparation independence, or accounts not representing preparations as uncertainty over observer-independent states of this specified kind, are outside this conclusion. Epistemic uncertainty between mixed preparations is not banned. Nor does the theorem establish a collapse mechanism, choose between Bohmian and Everettian theories, or solve the measurement problem. Its achievement is a conditional impossibility theorem about a precise form of pure-state epistemic overlap.