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Do ψ-ontic models capture the reality of quantum states? A measure-theoretic response.

Doyeong Kim

Published Dec 22, 2025• 0 citations
Publisher

Abstract

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).

DOI
10.11575/prism/50931

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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).
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.
Ψ-epistemic models of quantum mechanics imply that the quantum state does not correspond to physical reality, but instead reflects the observer's knowledge of the underlying quantum system. The epistemic view of the quantum state has the potential to shed light on several foundational problems of quantum theory and has attracted considerable attention in the literature. On the other hand, the Pusey-Barrett-Rudolph theorem demonstrated that broad classes of ψ-epistemic models must lead to predictions that deviate from those of quantum mechanics. Although the original theorem involved entangled joint measurements on composite systems, alternative no-go theorems involving measurements on single quantum systems were developed shortly thereafter. Experimental investigations of the deviations predicted by ψ-epistemic models from quantum mechanics are still ongoing. So far, such tests have been performed within the framework of non-relativistic quantum mechanics and predominantly rely on quantum information based measurement procedures. In this work, we show that ψ-epistemic models can give rise to deviations from standard quantum field-theoretic predictions through modifications of polarized scattering cross sections and decay widths. Our results do not require a relativistic formulation of ontological models or of the Harrigan-Spekkens criterion; the essential assumption is merely that measurements implemented through relativistic processes can still be represented within the ontological framework by well-defined response functions and probabilities. The present work constitutes a proof-of-principle study demonstrating that particle physics tests of the ontological status of the quantum state are possible and that ψ-epistemic models may exhibit experimentally distinguishable signatures in particle phenomenology.
The main question under investigation in this thesis was the reality of the quantum state. We approached this problem by studying a recent no-go result by Pusey, Barrett and Rudolph, namely the PBR theorem. In Section 1 we provided a short introduction to quantum theory. A few relevant examples were covered, namely decompositions of quantum states, impossibility of the ignorance interpretation and the distinguishability of pure quantum states. Section 2 developed measure theoretic tools needed in the framework of ontological models. The Radon-Nikodym theorem was proven in order to justify the use of an alternative expression of the variational distance of probability measures. In Section 3 we developed a general framework for ontological models. In short, the ontological models framework can be considered the most general framework currently available for hidden variable theories and no-go results. We used the measure theoretic tools of Section 2 to make the formalism rigorous. Original research included in Section 3 was the characterization of a property called antidistinguishability for qubit states. A necessary and sufficient condition for antidistinguishability was given. It was then proven that any set of pure qubit states that is not antidistinguishable can be made antidistinguishable by adding a single carefully chosen state into the set. In the end of Section 3, some original observations were presented. Namely, a specific connection with contextuality was proven for sometimes ψ-ontic and never ψ-ontic ontological models. However, it was then noted that these observations were not enough to exclude never ψ-ontic ontological models, a feat that would have provided a new no-go theorem on its own. Contextuality in the ontological models framework was studied in Section 4. The main results were establishing preparation contextuality from the failure of maximal ψ-epistemicity and a proof of Bell's theorem based on preparation contextuality established this way. We concluded that ψ-ontic models cannot be Bell local based on these results. Finally in Section 5 the implications of ψ-ontology were discussed. By careful analysis of the results, it was noted that accepting the reality of the quantum state exposes explanatory gaps in the formalism. Possible solutions to the challenges introduced by these explanatory gaps were discussed through interpretations of quantum mechanics and the so-called measurement problem.
In discussion of the interpretation of quantum mechanics the terms ‘ontic’ and ‘epistemic’ are often used in the sense of pertaining to what exists, and pertaining to cognition or knowledge respectively. The terms are also often associated with the formal definitions given by Harrigan and Spekkens for the wavefunction in quantum mechanics to be �ψ-ontic or �ψ-epistemic in the context of the ontological models framework. The formal definitions are contradictories, so that the wavefunction can be either ψ�-epistemic or ψ �-ontic but not both. However, we argue, nothing about the informal ideas of epistemic and ontic interpretations rules out wavefunctions representing both reality and knowledge. The implications of the Pusey–Barrett–Rudolph theorem and many other issues may be rethought in the light of our analysis.
Abstract In discussion of the interpretation of quantum mechanics the terms ‘ontic’ and ‘epistemic’ are often used in the sense of pertaining to what exists, and pertaining to cognition or knowledge respectively. The terms are also often associated with the formal definitions given by Harrigan and Spekkens for the wavefunction in quantum mechanics to be ψ\psi ψ ψ -ontic or ψ\psi ψ ψ -epistemic in the context of the ontological models framework. The formal definitions are contradictories, so that the wavefunction can be either ψ\psi ψ ψ -epistemic or ψ\psi ψ ψ -ontic but not both. However, we argue, nothing about the informal ideas of epistemic and ontic interpretations rules out wavefunctions representing both reality and knowledge. The implications of the Pusey–Barrett–Rudolph theorem and many other issues may be rethought in the light of our analysis.
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.