PBR Theorem

The Pusey–Barrett–Rudolph (PBR) theorem constrains models in which a preparation produces an underlying physical state and measurements read that state probabilistically. If independently prepared systems have independent underlying states, distinct pure quantum states cannot correspond to overlapping underlying probability measures while reproducing all quantum predictions. The theorem does not say that every interpretation must regard the wavefunction as a material field, and preparation independence is an essential premise.

Helpful background. Born Rule supplies probabilities for the joint measurement; Hidden-Variable Models defines ontic preparation measures and response functions.

Pure states and ontic overlap

Write an ontic state as $\lambda\in\Lambda$ and the probability measure assigned to preparation $\psi$ as $\mu_\psi$. For a fixed measurement $M$, its response functions obey

$$\xi(k\mid M,\lambda)\ge0,\qquad \sum_k\xi(k\mid M,\lambda)=1.$$

The model reproduces the Born rule when integrating each response against $\mu_\psi$ gives the quantum outcome probability. The response may depend on the full measurement context; PBR does not assume measurement noncontextuality.

Two preparation measures overlap when they share a nonzero common submeasure. With densities relative to a common dominating measure, its maximal mass is

$$\omega(\psi,\phi)=\int\min\{\mu_\psi(\lambda),\mu_\phi(\lambda)\}\,d\lambda.$$

This statistical overlap is distinct from the Hilbert-space overlap $|\langle\psi|\phi\rangle|$. Nonorthogonal vectors can have disjoint ontic measures: their measurement responses then fail to reveal the complete ontic state. Ontic distinctness does not imply that quantum measurements can distinguish the preparations perfectly.

The terminology $\psi$-ontic means pairwise disjointness for distinct pure states in this model framework. A theory may still include variables besides $\psi$. The broader ontic/epistemic model discussion separates that technical classification from ordinary statements about knowledge.

Preparation independence supplies the crucial composition rule

For separately chosen preparations $\psi_{x_1},\ldots,\psi_{x_n}$, the relevant assumption is

$$\mu_{x_1\cdots x_n} =\mu_{x_1}\otimes\cdots\otimes\mu_{x_n}$$

on a product ontic space. It includes a composition assumption and absence of relevant correlations between independently prepared components. Observing a product density operator alone does not establish this underlying factorization. It is also different from Bell’s independence of measurement settings and hidden variables.

Suppose two single-system preparations share common probability mass $\omega>0$. Product preparation independence makes every length-$n$ preparation string share a common component of mass $\omega^n$. A joint measurement cannot know which of those preparation labels was used when the underlying joint state lies in this common component.

PBR construct a quantum measurement for which each outcome has probability zero for one of the possible strings. Response normalization makes those exclusions incompatible with a common component: every possible outcome is forbidden by some preparation that can reach it. The full two-copy proof gives the orthonormal exclusion basis for $|0\rangle$ and $|+\rangle$, including measure-theoretic qualifications. More closely overlapping pairs require an appropriate larger-copy construction.

A noise calculation and its limits

Let $\varepsilon_x$ be the observed-model probability of the nominally forbidden outcome for string $x$. The common-component argument yields

$$\omega^n\le\sum_x\varepsilon_x.$$

For the four preparations in the two-copy example, suppose each excluded probability is bounded by $0.0025$. Then $\omega^2\le0.01$, hence $\omega\le0.1$. This is a bound on overlap within the assumed model, not a percentage probability that the wavefunction is real. Finite experimental counts require confidence bounds before frequencies can replace probabilities. Correlated preparation errors can invalidate product independence even when all four outcome frequencies are small.

What the theorem leaves undecided

PBR does not establish a preferred ontology, collapse, a physical-space wave field, or an interpretation of probability. It also does not exclude hidden-variable models with $\psi$ among their variables. It restricts a declared model class. Rejecting preparation independence changes that class and incurs an explanatory obligation: state how composition works and how independently controlled devices acquire the correlations required by the model.

Operational or agent-centered accounts need not posit preparation measures over an observer-independent ontic space in the first place. Applying the theorem to them requires an explicit translation into its framework, not merely noticing that they call the quantum state information.

Exercise: a product state is not an ontic product law

Let two hidden bits have uniform marginals but satisfy $\lambda_1=\lambda_2$ with certainty. Compare their joint probability of $(0,0)$ with the independent case. Explain the relevance to PBR.

Solution

The correlated model gives $p(0,0)=1/2$, whereas independent uniform bits give $1/4$. Marginal preparation tests cannot distinguish these hidden laws. PBR assumes the product law at the ontic level; correct marginals, or a stipulated product quantum state, do not by themselves prove that law.

References

• Harrigan, Nicholas, and Robert W. Spekkens, “Einstein, Incompleteness, and the Epistemic View of Quantum States,” Foundations of Physics 40, 125–157 (2010), doi:10.1007/s10701-009-9347-0 — the ontological-model classification.
• Leifer, Matthew S., “Is the Quantum State Real? An Extended Review of $\psi$-Ontology Theorems,” Quanta 3, 67–155 (2014), doi:10.12743/quanta.v3i1.22 — interpretation and composition assumptions.
• Pusey, Matthew F., Jonathan Barrett, and Terry Rudolph, “On the reality of the quantum state,” Nature Physics 8, 475–478 (2012), doi:10.1038/nphys2309 — the original many-copy theorem.