= Solution
\b[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.<Einstein, incompleteness, and the epistemic view of quantum states by Nicholas Harrigan and Robert W. Spekkens>[classification of ontological models and the distinction between epistemic overlap and ontic incompleteness]{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 assigns
$$
\mu_{x_1\ldots x_n}(\lambda_1,\ldots,\lambda_n)
=\prod_{k=1}^n\mu_{x_k}(\lambda_k).
$$
Operationally independent devices do not logically entail this ontic factorization; it is an additional physical premise.<On the reality of the quantum state by Matthew F. Pusey and others>[arXiv:1111.3328v3, pp. 2–4, equations (1)–(5): preparation measures, independent composition and normalized measurement response]{ref}
The two-<qubit> example can be checked directly. Set $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt2$. In <computational basis> order $00,01,10,11$, take the following normalized columns as an <orthonormal basis>:
$$
\xi_1=\frac{(0,1,1,0)^T}{\sqrt2},\quad
\xi_2=\frac{(1,-1,1,1)^T}{2},\quad
\xi_3=\frac{(1,1,-1,1)^T}{2},\quad
\xi_4=\frac{(1,0,0,-1)^T}{\sqrt2}.
$$
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 force
$$
\int\xi_M(j|\lambda_1,\lambda_2)\,d(\nu\otimes\nu)=0
\quad(j=1,2,3,4).
$$
Summing 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 $|\psi_{0,1}\rangle=\cos(\theta/2)|0\rangle\pm\sin(\theta/2)|1\rangle$, with overlap $\cos\theta$ and $0<\theta<\pi/2$. The general proof constructs an exclusion measurement on $n$ independently prepared copies when $2^{1/n}-1\leq\tan(\theta/2)$. 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.<On the reality of the quantum state by Matthew F. Pusey and others>[pp. 3–4 and Appendices A–B: finite-copy condition and noise-robust bound]{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.
Back to article page
