From Joint to Single-System Psi-Onticity Without Preparation Independence

Abstract: The Pusey-Barrett-Rudolph (PBR) theorem establishes $ψ$-onticity for individual quantum systems, but its standard formulation relies on the Preparation Independence Postulate (PIP). This has led to a prevalent view that rejecting PIP leaves open the possibility of $ψ$-epistemic models for individual systems. In this work, we show that this understanding is incomplete: once the PBR theorem establishes $ψ$-onticity for composite systems prepared in product states, the $ψ$-onticity of the individual subsystems follows directly from the tensor-product structure of quantum mechanics, without invoking PIP or any further auxiliary assumptions. This result removes a key auxiliary assumption from the PBR theorem, closes a persistent loophole for preserving $ψ$-epistemic models, and strengthens the conceptual foundations of $ψ$-ontology.

• The paper establishes that joint ψ-onticity implies individual subsystems are ψ-ontic without requiring Preparation Independence.
• It rigorously derives subsystem ψ-onticity from the tensor-product structure of composite quantum states, resolving a key conceptual loophole in ψ-ontology theorems.
• The analysis critiques alternative ψ-epistemic models, demonstrating that correlated ontic states cannot consistently reproduce quantum predictions for entangled measurements.

From Joint to Single-System $\psi$-Onticity Without Preparation Independence

Overview

This work addresses a central issue in the foundations of quantum mechanics: whether the quantum state ($\psi$) corresponds to an intrinsic physical property of an individual system ($\psi$-onticity), as opposed to representing mere knowledge about the system ($\psi$-epistemicity). The Pusey–Barrett–Rudolph (PBR) theorem is the cornerstone result in this debate, but its traditional derivation relies on the Preparation Independence Postulate (PIP). The accepted position has been that rejecting PIP leaves open $\psi$-epistemic models at the single-system level, even though joint $\psi$-onticity holds for composite systems. The paper rigorously demonstrates that once joint $\psi$-onticity is established for product states, the $\psi$-onticity of subsystems follows directly from the tensor-product structure of quantum theory, independent of PIP or any further auxiliary assumptions. This result resolves a longstanding conceptual loophole in the interpretation of $\psi$-ontology theorems.

$\psi$-Onticity, $\psi$-Epistemicity, and the PBR Theorem

Within ontological models, $\psi$-onticity corresponds to the property that distinct quantum states yield epistemic distributions supported on disjoint sets in the ontic space, i.e., an ontic state identifies a unique quantum state. Otherwise, the theory is $\psi$-epistemic, admitting the possibility that the same ontic state is compatible with multiple quantum state preparations.

The PBR theorem establishes that, under PIP, if the quantum statistics for entangled measurements are respected, then the supports of epistemic distributions for product states are disjoint: the joint state is $\psi$-ontic. However, the move from joint to single-system $\psi$-onticity in the PBR proof explicitly uses PIP, which asserts that independently prepared systems have uncorrelated ontic states. If PIP is relinquished, the literature claims (e.g., [Leifer 2014]) that single-system $\psi$-epistemicity remains an open possibility.

Structural Consequence of Joint $\psi$-Onticity

The key argument advanced is that $\psi$-onticity at the joint (composite) level implies that the epistemic distribution for a product state $|\psi_1\rangle \otimes |\psi_2\rangle$ takes the structural form

$$\mu_{\psi_1 \otimes \psi_2}(\lambda) = \delta(\lambda_\psi - \psi_1 \otimes \psi_2)\, \nu_{\psi_1\otimes\psi_2}(\eta)$$

where the ontic state is decomposed as $\lambda = (\lambda_\psi, \eta)$, with $\lambda_\psi$ specifying the prepared quantum state and $\eta$ parametrizing any residual (possibly highly correlated) hidden variables. Crucially, this decomposition follows rigorously from the disjointness of supports—no further independence or factorization assumption is needed.

Thanks to the tensor-product structure of quantum states, $\lambda_\psi$ can itself be decomposed canonically into $(\lambda_{\psi_1}, \lambda_{\psi_2})$, with each $\lambda_{\psi_i}$ corresponding to subsystem $i$. Disjointness at the composite level forces $\lambda_{\psi_1}$ and $\lambda_{\psi_2}$ to be respectively fixed by $\psi_1$ and $\psi_2$. The marginal epistemic distributions for the subsystems are therefore

$$\mu_{\psi_1}(\lambda_{\psi_1}) = \delta(\lambda_{\psi_1} - \psi_1), \qquad \mu_{\psi_2}(\lambda_{\psi_2}) = \delta(\lambda_{\psi_2} - \psi_2)$$

which are manifestly disjoint for distinct states. Thus, each subsystem is individually $\psi$-ontic, and the conclusion does not rely on independence of subsystems or the statistical separability of the ontic space.

On Proposed $\psi$-Epistemic Models

Models seeking to maintain single-system $\psi$-epistemicity by relaxing PIP, notably those of Lewis et al. [Lewis 2012] and Aaronson et al. [Aaronson 2013], are critically analyzed. These models construct correlated ontic states, allowing distinct quantum states of single systems to share ontic states. However, they do not supply a consistent framework for the composite ontic state space or provide response functions sufficient to reproduce quantum mechanics’ predictions for entanglement measurements, which are central to the PBR argument. Therefore, these models do not avoid the structural consequences of composite $\psi$-onticity, and their purported loophole is closed under the current proof.

Correlations and the Scope of Hidden Variables

The analysis permits arbitrarily strong correlations among hidden variables in the composite ontic space via $\nu_{\psi_1 \otimes \psi_2}(\eta)$. Nonetheless, such correlations cannot “wash out” the sharp $\psi$-label imposed by joint $\psi$-onticity, since the support for the wavefunction component is strictly on $\psi_1$ (for system 1) or $\psi_2$ (for system 2) and disjoint for alternatives. The argument shows that the structure of quantum theory enforces $\psi$-onticity locally in subsystems once it is present globally in product states.

Implications and Theoretical Significance

This result elevates the tensor-product structure of quantum theory, combined with experimentally verifiable joint $\psi$-onticity, to a sufficient foundation for ruling out single-system $\psi$-epistemicity—even in the strongest versions where PIP is entirely dispensed with. As such, it identifies the core quantum structure making the quantum state an intrinsic property of individual systems rather than a property only of composites or ensembles.

The approach clarifies that any claim that $\psi$-epistemic models are still viable (provided PIP is denied) implicitly ignores the full implications of $\psi$-onticity at the joint level or concedes that quantum predictions for entanglement measurements are not reproduced.

Conclusion

Joint $\psi$-onticity for product states, as established by the PBR theorem and enforced by the tensor-product structure of quantum mechanics, necessarily implies single-system $\psi$-onticity. Preparation Independence is not required at any stage, invalidating a commonly held belief in the $\psi$-ontology literature. The argument isolates $\psi$-onticity as a necessary structural feature of ontological models reproducing quantum theory, and confirms the quantum state as a categorical property of the individual system.

References:

• "Distinct quantum states can be compatible with a single state of reality" [Lewis et al., 2012, Phys. Rev. Lett. 109, 150404]
• "$\psi$-epistemic theories: The role of symmetry" [Aaronson et al., 2013, Phys. Rev. A 88, 032111]
• "Is the quantum state real? An extended review of $\psi$-ontology theorems" [Leifer, 2014, (Leifer, 2014)]
• "On the reality of the quantum state" [PBR, 2012, Nature Phys. 8, 475]

ArXiv ID: (2601.17662)