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the earlier claim that global symplectic non-factorisation obstructs PBR preparation independence; that inference is withdrawn.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 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 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 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 is widely regarded as one of the most important results in the foundations of quantum mechanics. It shows that, under broad and physically reasonable assumptions, the quantum wave function cannot be interpreted as a purely epistemic object. This conclusion has traditionally been taken to force a binary choice: either accept the wave function as ontic, or reject one of the theorem’s assumptions. This work presents a third option. Triadic Mesh Dynamics (TMD) provides an ontological framework in which the wave function ψ is not a fundamental physical entity, but an emergent statistical quantity summarizing the behaviour of a deeper, deterministic, and local triadic dynamics. In TMD, the ontic objects are the orientations of triads and their local flip dynamics, while ψ arises as a macroscopic description of orientational statistics, analogous to temperature in thermodynamics. Within this framework, distinct wave functions correspond to distinct statistical configurations of the triadic network, and TMD therefore automatically satisfies the empirical conclusion of the PBR theorem—without requiring ψ to be ontic. The result is a consistent third path: ψ is neither ontic nor epistemic, but emergent. This paper outlines the formal mapping between triadic orientation statistics and the emergent amplitude ψ, shows how the Schrödinger equation arises as the continuous limit of flip‑field dynamics, and argues that the PBR theorem naturally supports—rather than excludes—emergent ontologies such as TMD.
This paper contains initial work on attempting to bring recent developments in the foundations of quantum mechanics concerning the nature of the wavefunction within the scope of more logical and structural methods. A first step involves dualising a criterion for the reality of the wavefunction proposed by Harrigan & Spekkens, which was central to the Pusey-Barrett-Rudolph theorem. The resulting criterion has several advantages, including the avoidance of certain technical difficulties relating to sets of measure zero. By considering the 'reality' not of the wavefunction but of the observable properties of any ontological physical theory a new characterisation of non-locality and contextuality is found. Secondly, a careful analysis of preparation independence, one of the key assumptions of the PBR theorem, leads to a precise analogy with the kind of locality prohibited by Bell's theorem. Motivated by this, we propose a weakening of the assumption to something analogous to no-signalling. This amounts to allowing global or non-local correlations in the joint ontic state, which nevertheless do not allow for superluminal signalling. This is, at least, consistent with the Bell and Kochen-Specker theorems. We find a counter-example to the PBR argument, which violates preparation independence, but does satisfy this physically motivated assumption. The question of whether the PBR result can be strengthened to hold under the relaxed assumption is therefore posed.
Ontological models of quantum theory seek to realize quantum theory as a statistical theory over underlying degrees of freedom, whose measurement statistics are described by classical probability theory. Harrigan and Spekkens (2010) proposed a distinction between ontological models in which pure quantum states represent states of reality (ψ-ontic) or knowledge about reality (ψ-epistemic) depending on whether their associated probability distributions overlap, subsequently formalized into measure-theoretic form by Leifer (2014). In this thesis, I examine whether this ψ-ontic/epistemic distinction truly captures the intended difference between ontic and epistemic conceptions of pure states. I consider some mathematical alternatives relating to different notions of non-overlap between probability measures and compare them. I discuss the implications of these results for quantum ontology theorems – theorems that rule out ψ-epistemic readings of pure states under certain assumptions about the ontological model – such as the Pusey-Barrett-Rudolph Theorem (2012).