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

One of the most profound questions in physics is whether the quantum state, the wave function, represents something physically real or merely encodes an observer’s incomplete knowledge about a system. For more than a decade, the definitive answer seemed to rest on a landmark 2012 result known as the Pusey–Barrett–Rudolph theorem, which argued that the wave function must be real. But that celebrated proof carried an asterisk: it leaned on a contested auxiliary assumption called the Preparation Independence Postulate, leaving a lingering loophole through which defenders of the epistemic view, the idea that the quantum state is nothing more than information, could retreat. Now, in a paper published in the journal Foundations of Physics, physicist Shan Gao of the Research Center for Philosophy of Science and Technology at Shanxi University in Taiyuan, China, has closed that loophole. His work shows that once the wave function is established as real for composite systems, its reality for individual systems follows automatically from the mathematical structure of quantum mechanics itself, with no need for preparation independence or any additional assumptions.

The debate over the reality of the quantum state goes back to the very foundations of quantum theory in the 1930s. In the ontological models framework, formalized by Nicholas Harrigan and Robert Spekkens in 2010, a quantum system is assumed to possess underlying physical properties, described by an ontic state often denoted lambda, and the quantum state determines a probability distribution over these ontic states. If two distinct quantum states correspond to probability distributions that can overlap, sharing some common ontic states, then the quantum state is epistemic: it is a state of knowledge rather than a state of the world. If, however, the distributions for distinct quantum states never overlap, each ontic state is compatible with only one quantum state, and the wave function is ontic, a genuine physical property of the individual system.

The PBR theorem, published in Nature Physics in 2012 by Matthew Pusey, Jonathan Barrett, and Terry Rudolph, was designed to settle this question. The argument considers a composite system consisting of several subsystems, each prepared independently in one of two distinct quantum states. The theorem proves that if the quantum state were epistemic, the overlapping distributions for the different preparations would allow measurement statistics that contradict quantum mechanics, with a nonzero probability of obtaining outcomes that quantum theory forbids. The conclusion was striking: in any ontological model reproducing quantum predictions, the probability distributions associated with distinct quantum states must have disjoint supports in the ontic space. The wave function, in other words, is real.

Yet from the beginning, physicists noticed a subtlety. The PBR experiment, as the thought experiment is sometimes called, is performed on a composite system, a collection of subsystems prepared in product states. To move from the conclusion that the joint quantum state of the composite system is real to the claim that the wave function of each individual subsystem is real, the original argument invoked the Preparation Independence Postulate. This postulate states that independently prepared systems have independent ontic states, meaning the joint probability distribution over ontic states factors into a product of distributions for the subsystems. Critics, including Peter Lewis and his colleagues, as well as Samuel Aaronson and collaborators, pointed out that this is a nontrivial assumption, and moreover one that sits uncomfortably with the phenomenon of quantum entanglement, which the PBR framework itself takes seriously. If preparation independence is rejected, it seemed, the psi-epistemic position could survive, at least for individual systems. This view became widespread in the foundations-of-physics community, shaping how the theorem was taught and interpreted for over a decade.

Gao’s new paper demonstrates that this widespread understanding is incomplete, and in a rather elegant way. The key insight is that the tensor-product structure of quantum mechanics already contains everything needed to propagate onticity from the composite system to its parts. In quantum theory, the Hilbert space of a composite system is built as the tensor product of the Hilbert spaces of its subsystems, and the product state of a composite system is precisely the tensor product of the states of its components. Once the PBR theorem establishes that the composite-system wave function is ontic, every ontic state of the composite carries a sharp label identifying the prepared product state. But that label has the form psi-one tensor psi-two, a structured object whose components are the individual subsystem wave functions. Since the composite ontic state determines the full product state, it necessarily determines each subsystem state. The onticity of the parts follows from the onticity of the whole.

To make this argument rigorous, Gao supplies a mathematical representation theorem for psi-ontic models, presented in the appendix of the paper. The theorem considers a family of probability measures on an ontic space, one for each quantum state, with pairwise disjoint supports, which is exactly the operational content of psi-onticity. It proves that there exists a measurable function mapping each ontic state to the unique quantum state whose support contains it, allowing the ontic state to be decomposed into a component that equals the quantum state itself and a set of residual degrees of freedom. In this representation, the preparation distribution takes a delta-function form over the quantum-state component, showing that the familiar delta-function models used in such arguments are not extra assumptions but direct mathematical consequences of psi-onticity. Applying this theorem to the family of product-state preparations establishes that every ontic state of the composite system carries a uniquely determined label that exactly identifies the prepared product state, providing a precise structural foundation for the subsequent derivation of subsystem onticity.

The result is significant because it removes a key auxiliary assumption from the PBR theorem and closes what had been regarded as the main escape route for psi-epistemic models. As Gao emphasizes in the paper, the joint psi-onticity in the original PBR argument was actually derived entirely at the level of the composite system and never relied on preparation independence for the subsystems; that assumption entered only in the subsequent step inferring single-system onticity from joint onticity. His contribution is to show that this final step requires no assumption at all, because the tensor-product structure does the work. The question of whether the quantum state is real can no longer be dodged by rejecting preparation independence, at least within the ontological models framework, since psi-onticity for individual systems is now a theorem rather than a conditional conclusion.

The philosophical stakes of this clarification are considerable. The distinction between ontic and epistemic interpretations of the wave function lies at the heart of long-running debates about the meaning of quantum mechanics, echoing Einstein’s famous incompleteness objections. Mark Schlosshauer and Arthur Fine analyzed the implications of the PBR theorem in 2012, and in 2014 they also proved a no-go theorem for the composition of quantum systems, highlighting the delicate role of composition assumptions in this domain. Matthew Leifer’s extended 2014 review of psi-ontology theorems catalogued the landscape of results and loopholes, and the preparation independence loophole figured prominently among them. Gao, who has long worked on the ontology of quantum mechanics and authored the 2017 book The Meaning of the Wave Function, has now eliminated one of the most cited of these loopholes. What remains for the epistemic camp is essentially the rejection of the ontological models framework itself, or the embrace of exotic positions such as super-quantum nonlocality, rather than any escape resting on preparation independence.

For working physicists, the practical consequences may seem abstract, but the foundational clarity matters. Interpretations such as Bohmian mechanics and many-worlds, in which the wave function is physically real, emerge from this analysis unscathed and now rest on firmer logical ground. Epistemic reconstructions of quantum theory must confront the fact that the reality of the quantum state for individual systems follows from the reality of the quantum state for composite systems plus the tensor-product mathematics that all quantum theories share. In an era when quantum technologies are being built on precise control of multi-particle entangled states, knowing that the formalism’s central object describes something real, and knowing that this conclusion no longer depends on contestable independence assumptions, is more than philosophical housekeeping. It is a sharpened statement of what quantum mechanics is telling us about the world.

The paper, received in September 2025 and published in December 2025 in volume 56 of Foundations of Physics, is titled From Joint to Single-System psi-Onticity Without Preparation Independence. It stands as a reminder that even celebrated no-go theorems can be strengthened, decades-old loopholes can be sealed, and the deepest interpretive questions in physics can still yield to careful mathematical analysis.