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Outcome determinism says that a complete hidden state and the measurement settings fix each measurement outcome with probability one.

For the spin singlet state, measurements along the same axis are perfectly anticorrelated. By measuring either spin component of particle $B$, one can therefore predict with certainty the corresponding result for distant particle $A$. The Einstein–Podolsky–Rosen criterion of reality then says that every such spin component of $A$ is an element of physical reality, because the choice at $B$ can be made without disturbing $A$. Since the quantum state assigns no simultaneous sharp values to noncommuting spin components, the EPR argument concludes that the wavefunction is not a complete description, provided this locality premise is accepted.

Let $\lambda$ denote a proposed complete hidden state, and let $a,b$ be the chosen axes.

For two axes on each side, every $\lambda$ obeysMeasurement independence permits averaging the same distribution of $\lambda$ for all four setting pairs, producing the CHSH inequality

The singlet prediction isChoose coplanar unit vectorsThenand thereforeThus the predictions of quantum theory violate the conjunction of outcome determinism, parameter independence, and measurement independence. This is Bell theorem; the calculation alone does not select which premise a deeper theory must abandon.

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.

Outcome determinism means that a complete hidden state $\lambda$, together with the chosen measurement settings, fixes each outcome. The observed probabilities can still arise from an ensemble of unknown $\lambda$. This differs from deterministic unitary evolution of a wavefunction, which does not by itself specify a unique measurement outcome. Quantum nonlocality means failure of a Bell-local account of the correlations, under assumptions including measurement independence; it is stronger than the existence of entanglement and does not imply controllable faster-than-light signaling.

Use the first ket for the Positron and the second for the electron. Before either final beam splitter, the first pair of splitters and annihilation produceThe photon state $|\gamma ⟩$ is orthogonal to the surviving charged-particle sector. Its probability is $1/4$, while the remaining sector has squared norm $3/4$. The states below retain that photon amplitude and are normalized over all trials, without conditioning on nonannihilation. For compactness write $|cc⟩=|c^{+}{c}^{-}⟩$, and similarly for $|cd⟩,|dc⟩,|dd⟩$.