Psi Ontic Psi Epistemic Models

In an ontological model, a preparation of $|\psi\rangle$ assigns a probability distribution $\mu_\psi(\lambda)$ over an underlying physical state $\lambda$. A model is $\psi$-ontic when distinct pure states have nonoverlapping ontic distributions up to measure-zero sets. It is $\psi$-epistemic when at least one pair overlaps. These technical definitions are narrower than saying that a wavefunction is “knowledge.”

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

Ontological models of preparations and measurements

A measurement $M$ has response functions $\xi(a\mid M,\lambda)$ satisfying positivity and normalization. Reproducing quantum statistics requires

$$\int d\lambda\,\mu_\psi(\lambda)\xi(a\mid M,\lambda) =\langle\psi|E_{a|M}|\psi\rangle.$$

The hidden state need not be deterministic: response functions can be probabilistic. Nor is $\lambda$ required to be a particle position. The framework specifies a representation of preparations and measurements; additional assumptions constrain composition and dynamics.

What overlap means

For distributions on a common measure space, a useful overlap quantity is

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

If $\omega>0$, some underlying states are compatible with either preparation. The quantum distinction between $|\psi\rangle$ and $|\phi\rangle$ then partly reflects uncertainty about $\lambda$.

Orthogonal states must have zero overlap in a model reproducing perfect discrimination. A measurement distinguishes them with certainty, so its response would have to be both 1 and 0 on any overlap of positive measure, a contradiction.

For nonorthogonal states, imperfect quantum discrimination alone does not determine whether overlap exists. A $\psi$-ontic model can still make some ontic information inaccessible.

PBR and its assumptions

The Pusey–Barrett–Rudolph theorem constrains overlapping distributions when independently prepared systems satisfy a preparation-independence assumption. The joint ontic distribution is then a product of the individual ones. Certain joint measurements lead to a contradiction if the overlaps persist.

This does not prove that every possible use of epistemic language is impossible. Rejecting the ontological-model framework, altering preparation independence, or changing other hypotheses changes the theorem’s applicability. Those changes have explanatory costs that must be assessed explicitly.

Ontology is not completeness

A $\psi$-ontic model need not say the wavefunction is the complete physical state. Bohmian mechanics contains a wavefunction and an actual configuration. Likewise, proving distinct wavefunctions correspond to disjoint ontic supports does not specify whether the wavefunction is a material field, a lawlike object, or something else.

QBism treats state assignments as personal expectations without necessarily introducing the hidden-variable space used here. Classifying it mechanically as $\psi$-epistemic in this technical sense can therefore be misleading.

Exercises

Show why perfect distinguishability forbids positive ontic overlap even if response functions are stochastic.

Solution

If outcome $a$ occurs with probability one for $\psi$, positivity implies $\xi(a|\lambda)=1$ almost everywhere under $\mu_\psi$. If it has probability zero for $\phi$, then $\xi(a|\lambda)=0$ almost everywhere under $\mu_\phi$. A region with positive overlap would require both values, which is impossible.

References

• M. S. Leifer, “Is the quantum state real? An extended review of $\psi$-ontology theorems,” Quanta 3, 67–155 (2014), author manuscript.
• M. F. Pusey, J. Barrett, and T. Rudolph, “On the reality of the quantum state,” Nature Physics 8, 475–478 (2012), doi:10.1038/nphys2309.