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arXiv:2601.17662 · Foundations of Physics

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

Shan Gao

quant-phSubmitted Jan 25, 2026• Announced Jan 27, 2026• 1 citation
arXivHTMLPDF

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.

Journal reference
Found Phys 56, 7 (2026)
DOI
10.1007/s10701-025-00910-w
Comments
12 pages, no figures

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The Pusey-Barrett-Rudolph (PBR) theorem establishes ψψψ-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 ψψψ-epistemic models for individual systems. In this work, we show that this understanding is incomplete: once the PBR theorem establishes ψψψ-onticity for composite systems prepared in product states, the ψψψ-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 ψψψ-epistemic models, and strengthens the conceptual foundations of ψψψ-ontology.
The Pusey-Barrett-Rudolph (PBR) theorem proves that the joint wave function ψ1 ⊗ψ2 of a composite quantum system is ψ-ontic, representing the system’s physical reality. We present a minimalist proof showing that this result, combined with the tensor product structure assigning ψ1 to subsystem 1 and ψ2 to subsystem 2, directly implies that ψ1 and ψ2 are ψ-ontic for their respective subsystems. This establishes ψ-ontology for single quantum systems without requiring preparation independence or other assumptions. Our proof challenges the widely held view that joint ψ-onticity permits subsystem ψ-epistemicity via correlations, providing a simpler, more direct understanding of the wave function’s ontological status in quantum mechanics.
The Pusey-Barrett-Rudolph (PBR) no-go theorem provides an argument for the reality of the quantum state by ruling out $\ensuremath{\psi}$-epistemic ontological theories, in which the quantum state is of a statistical nature. It applies under an assumption of preparation independence, the validity of which has been subject to debate. We propose two plausible and less restrictive alternatives: a weaker notion allowing for classical correlations, and an even weaker, physically motivated notion of independence, which merely prohibits the possibility of superluminal causal influences in the preparation process. The latter is a minimal requirement for enabling a reasonable treatment of subsystems in any theory. It is demonstrated by means of an explicit $\ensuremath{\psi}$-epistemic ontological model that the argument of PBR becomes invalid under the alternative notions of independence. As an intermediate step, we recover a result which is valid in the presence of classical correlations. Finally, we obtain a theorem which holds under the minimal requirement, approximating the result of PBR. For this, we consider experiments involving randomly sampled preparations and derive bounds on the degree of $\ensuremath{\psi}$ epistemicity that is consistent with the quantum-mechanical predictions. The approximation is exact in the limit as the sample space of preparations becomes infinite.
The Pusey–Barrett–Rudolph (PBR) theorem constrains ontological models in which distinct pure quantum states correspond to overlapping ontic preparation distributions and independently prepared systems satisfy preparation independence. Applying that result to Constraint-Surface Dynamics (CSD) requires care because CSD calls the projective quantum state epistemic in a different sense: the ray [ψ] is an incomplete, many-to-one image of an underlying ontic state x ∈ Σ. This paper separates these notions at the level of preparation measures. A general CSD preparation is a Borel probability measure on the ontic space Σ. We first prove the general measure-theoretic result that mutual singularity of two pushforward measures under a measurable map implies mutual singularity of the original measures. Hence distinct exact sharp preparations are mutually singular. The canonical exact CSD preparation interface is therefore ψ-ontic in the technical Harrigan–Spekkens sense and already satisfies the disjointness conclusion associated with the PBR theorem. We then distinguish this exact interface from finite-resolution region preparations. A region of positive Liouville measure has a conditional preparation law absolutely continuous with respect to Liouville measure, whose projective pushforward admits a Radon–Nikodym density with respect to Fubini–Study measure. On the concrete Kähler witness, individual projective fibres are Liouville-null while an exact sharp preparation assigns unit probability to its own fibre. Exact sharp measures are consequently singular with respect to Liouville measure, and no positive-volume region-conditioned preparation law can equal an exact sharp preparation law. Suitable finite-resolution region preparations may nevertheless overlap. These results explain how CSD can regard the quantum state as epistemic in the sense of being ontically incomplete while remaining ψ-ontic under the Harrigan–Spekkens classification of exact preparations. They also remove the need for the earlier claim that global symplectic non-factorisation obstructs PBR preparation independence; that inference is withdrawn. Preparation independence remains a separate compositional question. The preparation results used here are machine-checked in Lean 4. Programme background: https://www.constraintsurfacedynamics.com — Glossary: https://glossary.constraintsurfacedynamics.com
The Pusey-Barrett-Rudolph (PBR) theorem deals with the realism of the quantum state. It establishes that every pure quantum state is real, in the context of ψ\psiψ-ontic quantum ontological models. In order to achieve this, PBR consider an ad hoc postulate termed the Preparation Independence Postulate (PIP). Here we argue that even without the use of the PIP, the PBR argument leads to a link between scalability and realism of quantum states whereby the latter can be achieved through the former. Further, we consider a set of n-qubit registers in interaction with an environment by means of i) individual and ii) collective couplings. In both cases, we analytically show that the phenomenon of achieving realism through scalability is still present, even though this now occurs at a slower rate. Searching for an optimisation of this phenomenon, we report numerical experiments up to n = 4 qubits: for individual qubit-noise coupling, be it a bit, a phase, or a bit-phase flip noise, the numerical search exhibits the same response to all the different noisy channels, without improving the analytical expectation. In contrast, the collective qubit-noise coupling leads to a more efficient display of the phenomenon, and a variety of noise channel-dependent behaviours emerge.
We present a new psi-ontology theorem demonstrating that the quantum wave function is ontic (real) rather than epistemic (representing knowledge) in single-world unitary quantum theories (SUQTs). By leveraging a protocol of repeated reversible measurements on a single quantum system, we show that any two distinct quantum states produce different statistical distributions of (erased) measurement outcomes. This theoretical distinguishability implies that different quantum states correspond to different physical realities, supporting the ontic nature of the quantum state. Unlike previous psi-ontology theorems, such as the Pusey-Barrett-Rudolph theorem, our proof relies solely on the unitary evolution and Born rule of SUQTs, without additional assumptions like preparation independence. This strengthens its implications for quantum foundations, particularly in restricting non-psi-ontic interpretations like QBism without assuming an underlying ontic state and its dynamics. The theorem applies to any pair of distinct states in a finite-dimensional Hilbert space, with extensions to infinite-dimensional systems, offering a robust and general argument for the reality of the quantum state.
The Pusey--Barrett--Rudolph (PBR) theorem is commonly taken to establish that a pure quantum state represents a physical property of the system to which it is assigned, rather than merely encoding knowledge. In this work, I distinguish between state-first and dynamics-first physical theories, and argue that this reading presupposes the former. These two types of theories support different approaches to causal explanation, and one of PBR's stated assumptions depends on the state-first approach. Using a recent interpretation of quantum theory as a concrete example, I show how a dynamics-first theory can explain the PBR correlations while maintaining a broadly epistemic reading of quantum states. I conclude that the PBR theorem supports a dilemma: either the quantum state is part of reality, or reality is not fundamentally built out of states.