Is the quantum state real? An extended review of $ψ$-ontology theorems

M. S. Leifer

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Is the quantum state real? An extended review of $ψ$-ontology theorems

M. S. Leifer

arXiv:1409.1570v2Sep 4, 2014quant-ph

TL;DR

This review examines whether the quantum state must be real in realist approaches to quantum theory. It develops the background and presentation of the Pusey–Barrett–Rudolph result, surveys related arguments and criticisms, and concludes that ψ-epistemic explanations remain possible without auxiliary assumptions, though stronger versions may become implausible.

Problem
The review addresses whether the quantum state is an objective property of reality or instead a state of knowledge, a foundational question testable through rigorous argument and experiment.
Method
The review uses rigorous measure-theoretic probability and provides background, presentation, criticism, and implications for ψ-ontology results and related work.
Results
The Pusey–Barrett–Rudolph Theorem shows that ontological models reproducing quantum predictions and satisfying the PIP are ψ-ontic.
Takeaways & Limitations
Without auxiliary assumptions, ψ-epistemic models exist, while future bounds on measure overlap may make such explanations implausible within the ontological-models framework.
Takeaways & Limitations
The review notes that the NCA is not a Bell-locality assumption, and its reasonableness requires justification from quantum operational structure rather than Bell locality alone.

Abstract

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Towards the end of 2011, Pusey, Barrett and Rudolph derived a theorem that aimed to show that the quantum state must be ontic (a state of reality) in a broad class of realist approaches to quantum theory. This result attracted a lot of attention and controversy. The aim of this review article is to review the background to the Pusey--Barrett--Rudolph Theorem, to provide a clear presentation of the theorem itself, and to review related work that has appeared since the publication of the Pusey--Barrett--Rudolph paper. In particular, this review: Explains what it means for the quantum state to be ontic or epistemic (a state of knowledge); Reviews arguments for and against an ontic interpretation of the quantum state as they existed prior to the Pusey--Barrett--Rudolph Theorem; Explains why proving the reality of the quantum state is a very strong constraint on realist theories in that it would imply many of the known no-go theorems, such as Bell's Theorem and the need for an exponentially large ontic state space; Provides a comprehensive presentation of the Pusey--Barrett--Rudolph Theorem itself, along with subsequent improvements and criticisms of its assumptions; Reviews two other arguments for the reality of the quantum state: the first due to Hardy and the second due to Colbeck and Renner, and explains why their assumptions are less compelling than those of the Pusey--Barrett--Rudolph Theorem; Reviews subsequent work aimed at ruling out stronger notions of what it means for the quantum state to be epistemic and points out open questions in this area. The overall aim is not only to provide the background needed for the novice in this area to understand the current status, but also to discuss often overlooked subtleties that should be of interest to the experts.

1. Introduction

The review frames the reality of the quantum state as a foundational physics question and examines ψ-ontic and ψ-epistemic interpretations through the Pusey–Barrett–Rudolph theorem and related results.

Motivation: Bell’s theorem established quantum foundations as a field addressed through rigorous argument and experiment, rather than as metaphysics.The review rejects a sharp division between quantum theory’s practical content and its interpretation.
Subsequent work: The review also examines criticisms and auxiliary assumptions, then surveys stronger ψ-epistemic constraints and open questions.Without auxiliary assumptions, ψ-epistemic models exist, motivating further work on stronger notions of epistemicity.
Why ψ-ontology matters: Proving the wavefunction ontic would imply major no-go results, including Bell’s theorem and an ontic state space that is infinite and exponentially scaling.The review presents this as a potential unification of existing no-go theorems with possible relevance to quantum information theory.
Review structure: The review surveys prior arguments for both interpretations before presenting the Pusey–Barrett–Rudolph, Hardy, and Colbeck–Renner theorems.It gives the Pusey–Barrett–Rudolph theorem the most extensive treatment because it has generated the largest literature, confusion, and criticism.
The ψ-ontology question: The central question is whether a quantum state corresponds to an independently existing property or merely serves as a mathematical tool.This is the distinction between ψ-ontic and ψ-epistemic views.

2. Arguments for a ψ-epistemic interpretation

The review presents realist ψ-epistemic interpretations as approaches with an underlying ontology in which the wavefunction represents knowledge rather than reality. Spekkens’ toy theory illustrates how overlapping epistemic states can reproduce selected quantum phenomena and dissolve puzzles associated with an ontic wavefunction.

Realist ψ-epistemic interpretations posit an underlying ontology while treating the wavefunction as a representation of knowledge rather than part of reality.Unlike neo-Copenhagen interpretations, they retain a realist framework; ψ-ontology theorems therefore apply specifically to them.
2.1. Spekkens toy bit: Spekkens’ toy theory reproduces the qualitative behavior of spin-1/2 particles for preparations and measurements in the x, y, and z bases.The model is restricted here to a single toy bit, although the full theory includes dynamics, composite systems, and some entangled-state phenomena.
2.1. Spekkens toy bit: In Spekkens’ model, each quantum-like state corresponds to a probability distribution over ontic states, and nonorthogonal states can overlap on an ontic state.The overlap provides an epistemic explanation for why measurements cannot always distinguish which preparation occurred.
2.2. The no-cloning theorem: The toy model accounts for no-cloning because overlapping input distributions cannot be transformed into outputs with the same degree of overlap.The inputs overlap at 50%, whereas the corresponding outputs overlap only at 25%.
2.3. Mixed states: Different 50/50 mixtures of pure toy-bit states produce the same maximally mixed ontic distribution, explaining the non-uniqueness of mixed-state decompositions.This equivalence depends on overlap between the distributions associated with nonorthogonal states.
2.4. The collapse of the wavefunction: ψ-epistemic interpretations dissolve the measurement problem by allowing a superposition description to reflect ignorance about an underlying state that is already definite.The motivation contrasts this with the ontic view, where superposed and definite macroscopic states represent distinct physical situations.

3. Arguments for a ψ-ontic interpretation

Pre-theorem arguments for a ψ-ontic interpretation often rely on interference or eigenvalue-eigenstate reasoning, but realist ψ-epistemic models challenge both lines of argument. The review therefore treats these arguments as insufficient without further assumptions.

3. Arguments for a ψ-ontic interpretation: Arguments for ψ-ontology must distinguish proving that something is real from proving specifically that the wavefunction is real.The review notes that many earlier arguments overlook realist ψ-epistemic theories and thereby establish only the reality of some underlying entity.
3.1. Interference: Interference-based arguments contrast classical waves, which spread through both slits, with particles, which follow definite trajectories, to explain quantum interference and localized detection events.The review emphasizes that this classical dichotomy is not exhaustive because broader realist descriptions remain possible.
3.1. Interference: ψ-epistemic models can reproduce interference, including qualitative Mach–Zehnder experiments and delayed-choice and Elitzur–Vaidman bomb-test effects.Spekkens’ toy theory retains a fact about which interferometer arm the photon takes while reproducing these interference phenomena.
3.1. Interference: Because interference occurs in models reproducing quantum fragments exactly, inferring that the wavefunction is real requires additional assumptions.The review presents Hardy’s theorem as one attempt to add such assumptions, while questioning their plausibility.
3.1. Interference: The wavefunction-field analogy weakens for multiple particles because entanglement makes the quantum state space scale exponentially with system number.The wavefunction can no longer be viewed straightforwardly as a field on ordinary three-dimensional space.
3.2. The eigenvalue-eigenstate link: Spekkens’ toy theory shows that epistemic states can be uniquely associated with definite measurement values while remaining underdetermined by the underlying ontic state.For example, one ontic state can be compatible with several epistemic states.
3.2. The eigenvalue-eigenstate link: The eigenvalue-eigenstate argument fails when experimentally measurable observables and preparable states are not assumed to exhaust all properties and logically possible states.Without those assumptions, the argument is simply false.
3. Arguments for a ψ-ontic interpretation: Deutsch’s computational argument is presented as a challenge for ψ-epistemic interpretations rather than a decisive argument against them.The review notes that no viable ψ-epistemic interpretation covering all quantum theory had yet been constructed.

4. Formalizing the ψ-ontic/epistemic distinction

The review formalizes ψ-ontic and ψ-epistemic questions using ontological models of prepare-and-measure fragments. These models represent preparations by probability measures over ontic states and measurements by conditional outcome probabilities, while allowing contextuality and stochasticity.

4. Formalizing the ψ-ontic/epistemic distinction: Ontological models provide a formal framework for asking whether the quantum state is ontic or epistemic within a realist theory.The framework is related to Bell’s hidden-variable framework but is designed not to assume the wavefunction’s reality in advance.
4.1. Prepare and measure experiments: The Pusey–Barrett–Rudolph theorem uses prepare-and-measure experiments, whereas Hardy and Colbeck–Renner theorems also involve assumptions about dynamics.A prepare-and-measure experiment prepares a system, immediately measures it, records the outcome, and discards it.
4.1. Prepare and measure experiments: Prepare-and-measure experiments assume statistically independent repetitions and independent choices of preparation and measurement.The formalism allows repeated trials to generate frequency statistics for comparison with theoretical probabilities.
4.1. Prepare and measure experiments: A prepare-and-measure fragment F = ⟨H, P, M⟩ specifies a Hilbert space, a set of density operators, and a set of POVMs whose probabilities follow the Born rule.The review considers the most general quantum states and finite-outcome POVMs, while often specializing to pure states and complete orthonormal-basis measurements.
4.2. Ontological models: An ontological model introduces a measurable ontic state space whose elements provide a complete specification of the system’s properties as they exist in reality.A preparation may generate a probability measure over ontic states because it need not completely control which ontic state occurs.
4.2. Ontological models: Different preparation procedures for the same quantum state may induce different ontic probability measures, a phenomenon called preparation contextuality.This is especially relevant for mixed states because they can have multiple convex decompositions into pure states.
4.2. Ontological models: Measurements are represented by conditional probability distributions over outcomes, which may be stochastic and must be nonnegative, normalized, and measurable in the ontic state.For a two-outcome measurement, the outcome probabilities sum to one for every ontic state; predicted observable probabilities are obtained by averaging over the preparation measure.
4.2. Ontological models: Measurement contextuality means that distinct implementations of the same measurement may yield different conditional probability distributions, so a measurement is associated with a set of such distributions.The review notes that this contextuality is required in certain models by the Kochen–Specker theorem.

5. Implications of ψ-ontology

ψ-ontology is presented as a powerful constraint because it implies major no-go results, including Bell nonlocality and excess-baggage bounds, while weaker assumptions connect contextuality and preparation contextuality to similar conclusions.

Ontic-state-space implications: Hardy’s and Montina’s excess-baggage results establish, respectively, infinitely many ontic states for a qubit and exponential parameter growth with system size.These results provide prior examples of lower bounds on ontic-state-space size.
Ontic-state-space implications: ψ-ontology would imply Bell’s Theorem and excess-baggage results requiring an uncountably infinite ontic state space with exponentially many parameters.This follows because a ψ-ontic model must contain at least as many ontic states as quantum states.
Contextuality implications: Maximally ψ-epistemic models are excluded by ψ-ontology and, independently, by Kochen–Specker contextuality.The hierarchy therefore links non-maximal ψ-epistemicity to either ψ-ontology or Kochen–Specker contextuality.
Preparation contextuality: For a PM fragment containing nonorthogonal pure states, any reproducing maximally ψ-epistemic model is ψ-epistemic; with convexity, non-maximal ψ-epistemicity forces preparation contextuality.The latter result applies to the maximally mixed state under the stated conditions.
Bell’s Theorem: The resulting preparation contextuality implies Bell’s conclusion: a ψ-ontic model reproducing the relevant product-fragment predictions cannot be Bell local.Bell locality would induce a preparation-noncontextual conditional model respecting convexity, contradicting the preceding constraint.
Bell’s Theorem: The same Bell conclusion can alternatively be obtained from Kochen–Specker contextuality, but this route is more complicated than standard proofs.The review compares these implications rather than presenting the indirect route as the preferred derivation.

6. Antidistinguishability

Antidistinguishability generalizes distinguishability to sets of states by identifying which state was definitely not prepared. The review introduces n-way measure overlap to formalize its ontological consequences, including perfect exclusion when the overlap vanishes.

Concept and examples: Antidistinguishability uses a measurement whose outcome rules out a specific prepared state, unlike distinguishability, whose outcome identifies the prepared state.For three or more states, antidistinguishability is strictly weaker because nonorthogonal states can be antidistinguished.
Concept and examples: Figure 11 constructs an antidistinguishing POVM for three nonorthogonal equatorial states by scaling projectors onto orthogonal companion states by 2/3.The orthogonality relations ensure that the scaled projectors form the required POVM.
Overlap formalism: Unlike distinguishability, antidistinguishability constrains n-way overlaps among probability measures rather than only pairwise overlaps.This motivates introducing a generalized overlap for finite or countable sets of measures.
Overlap formalism: The overlap L({µ_j}) is the optimal failure probability multiplied by n for guessing a state that was not prepared, and L=0 means perfect antidistinguishability.The operational interpretation assumes equal prior probabilities and allows deterministic or probabilistic strategies.
Overlap formalism: For two probability measures, the generalized overlap reduces to one minus their variational distance: L(µ,ν)=1−D(µ,ν).Thus the n-way construction extends the familiar two-measure notion of distinguishability.
Ontological consequence: Any reproducing ontological model assigns zero n-way overlap to every collection of measures representing an antidistinguishable set of quantum states.The proof uses the fact that an ontic state assigned positive probability by every measure would have to assign zero probability to every POVM outcome.
Terminology: The terms “PP incompatibility” and “conclusive exclusion” denote the same concept as antidistinguishability in related literature.The review notes that these alternative names arose before and after the concept’s use in the Pusey–Barrett–Rudolph theorem.

7. The Pusey–Barrett–Rudolph Theorem

The Pusey–Barrett–Rudolph Theorem shows that, under the PIP, ontological models reproducing quantum predictions are ψ-ontic. The section also examines weakened assumptions, ψ-epistemic countermodels, and criticisms of alternative composition constraints.

7.1. The Preparation Independence Postulate: The PIP is the theorem’s most controversial assumption and receives detailed definition and criticism.The review notes that it appears superficially plausible but was not extensively discussed in the original Pusey–Barrett–Rudolph paper.
7.1. The Preparation Independence Postulate: Product models reproduce the quantum predictions for product states, while entangled states require additional ontic states and measures.The PIP applies only to product state fragments, so additional structure for entangled states does not violate it.
7.2. The main result: The Pusey–Barrett–Rudolph Theorem states that any model containing all pure states, reproducing quantum predictions, and satisfying the PIP is ψ-ontic.The proof uses antidistinguishability of suitable product-state sets to rule out the relevant overlaps between ontic distributions.
7.2. The main result: Antidistinguishability of four product states converts the absence of four-way overlap into ontological distinctness of the underlying single-system states.The example applies this reasoning to [0] and [+], whose product preparations cannot have four-way overlap.
7.3. Motivation for the PIP: The theorem presents a dilemma among giving up the CPA, the NPA, or ψ-epistemicism, with each option opening an explanatory gap.The review compares this dilemma with those associated with Bell’s Theorem and contextuality.
7.4. Weakened assumptions: Weakened assumptions can preserve the ψ-ontology conclusion, but their connection to operational properties of quantum theory becomes less clear.Under compactness and quantum preclusions, the generalized theorem still yields ψ-ontology.
7.5. Necessity of the PIP: ψ-epistemic models can make a single pair of pure states ontologically indistinct, but compatibility with composite systems is ruled out for some n by antidistinguishability.The construction therefore cannot remain compatible with the relevant product-state models on sufficiently large composites.
7.6.1. Criticism of the CPA: Replacing the PIP with the WPIP cannot establish ψ-ontology because ψ-epistemic models exist for the factor fragments and any pair of models satisfies the WPIP.The review nevertheless questions whether the WPIP is a viable constraint on subsystem composition.

8. Dynamics in ontological models

The review formalizes how ontological models represent transformations, including unitary dynamics and ancilla appending, and identifies preservation of ontological distinctness as the key dynamical constraint used in ψ-ontology arguments.

PMT fragments: PMT fragments extend prepare-measure theories with unitary transformations while requiring the preparation set to remain closed under those transformations.The fragment includes a Hilbert space, states, measurements, and unitaries containing the identity.
Ontological representation: Transformations are represented by Markov kernels, allowing stochastic transitions between ontic states rather than assuming deterministic dynamics.Different implementations of the same unitary may correspond to different kernels, making the model transformation contextual.
Ontological representation: A PMT ontological model supplements ontic-state, preparation, and measurement representations with transformation kernels constrained to map preparations of ρ into preparations of UρU†.The model may associate sets of measures, response functions, and kernels to account for preparation, measurement, and transformation contextuality.
Preservation of distinctness: Markov-kernel dynamics cannot increase variational distance between ontic distributions, so ontological distinctness is preserved under allowed unitary transformations.This yields the corollary that ontologically indistinct states remain indistinct after applying a unitary.
Preservation of distinctness: When all unitaries are available, whether two pure states are ontologically distinct depends only on their inner product.Unitary equivalence in both directions preserves the distinctness relation.
Appending ancillas: Appending an ancilla is modeled as a stochastic map between ontological models, and the same variational-distance argument makes it preserve ontological distinctness.This assumption enables extending finite-dimensional dynamical arguments to larger Hilbert spaces.

9. Hardy’s Theorem

Hardy’s theorem uses restricted ontic indifference and interferometric reasoning to establish ontological distinctness for sufficiently separated pure states, with ancilla extension yielding full ψ-ontology.

Ontic Indifference: Hardy’s theorem assumes ontic indifference: a unitary leaving a pure state invariant can be implemented without changing ontic states in that state’s support.Restricted ontic indifference requires this only for one pure state, rather than every pure state.
Ontic Indifference: Ontic indifference is controversial for ψ-epistemic models because transformations can permute ontic states within a state’s support, as in Spekkens’ toy theory.The review nevertheless notes a locality-based motivation for Hardy’s assumption.
An example: In the interferometer example, a phase shift leaves the localized state invariant but maps the superposition to a state with different certain detector behavior, producing a contradiction under ontic indifference.The argument concludes that the localized state and equal superposition cannot share an ontic state with nonzero probability.
The main result: For d≥3, restricted ontic indifference and quantum preclusions imply that pure states satisfying Tr(|φ⟩⟨φ| |ψ⟩⟨ψ|) ≤ (d−1)/d are ontologically distinct.The theorem applies to a PMT fragment with suitably chosen measurements and unitaries.
The main result: Appending sufficiently large ancillas preserves the relevant distinctness while increasing Hilbert-space dimension, allowing Hardy’s theorem to establish distinctness for every pair of pure states.The review uses this extension to conclude that the original model must be ψ-ontic.
The main result: The proof’s measure-theoretic step establishes positive overlap for the relevant family of measures, contradicting the theorem’s required zero-overlap condition.This contradiction forces the candidate states to be ontologically distinct.

10. The Colbeck–Renner Theorem

The Colbeck–Renner argument uses parameter independence and chained Bell measurements to derive ψ-ontology for arbitrary pairs of pure states under specified conditions.

10. The Colbeck–Renner Theorem: The theorem is based on parameter independence, a locality assumption weaker than Bell locality but sufficient to define unambiguous marginal models.Bell locality is equivalent to parameter independence together with outcome independence.
10. The Colbeck–Renner Theorem: The proof proceeds through marginal fragments, chained Bell measurements, and an equiprobability theorem for maximally entangled states.The equiprobability theorem states that relevant ontic states assign equal probabilities to outcomes of a local orthonormal-basis measurement.
10. The Colbeck–Renner Theorem: Parameter independence removes the ambiguity in deriving a marginal model because Bob’s measurement choice cannot affect Alice’s outcome probabilities.Without parameter independence, different choices of Bob’s measurement could yield different conditional distributions for Alice’s marginal fragment.
10. The Colbeck–Renner Theorem: The main theorem shows that, under parameter independence, unitary preservation of ontological distinctness, and quantum-prediction reproduction, arbitrary pairs of pure states are ontologically distinct.The result applies to product measurement fragments containing a maximally entangled state and extends the argument beyond one specific pair.
10. The Colbeck–Renner Theorem: The review judges parameter independence to be an auxiliary assumption whose status is less compelling than the central conclusion it supports.The review also notes that ψ-epistemic theories remain possible when auxiliary assumptions are not imposed.

11. Pairwise ψ-epistemic models

Pairwise ψ-epistemic models require every pair of nonorthogonal pure states to have overlapping ontic representations. The review describes constructions that realize this property and symmetry assumptions that can rule it out in restricted settings.

11. Pairwise ψ-epistemic models: A pairwise ψ-epistemic model makes every pair of nonorthogonal pure states ontologically indistinct.This strengthens the basic ψ-epistemic condition, which requires indistinctness for only one pair.
11. Pairwise ψ-epistemic models: ABCL construct pairwise ψ-epistemic models for all pure states and orthonormal-basis measurements in finite-dimensional Hilbert spaces.Their construction mixes models in which selected nonorthogonal pairs are ontologically indistinct.
11. Pairwise ψ-epistemic models: Mixing two ontological models places their ontic spaces side-by-side and preserves ontological indistinctness present in either component.The resulting mixture continues to reproduce the quantum predictions when both original models do so.
11. Pairwise ψ-epistemic models: Iterating the mixture construction can make any finite collection of nonorthogonal pairs ontologically indistinct, while extending it to all pairs requires more technical work.The infinite construction is described as technically involved but straightforward once the finite construction is understood.
11. Pairwise ψ-epistemic models: ABCL’s symmetry theorem rules out pairwise ψ-epistemic models for a restricted fragment with specified ontic spaces and unitary dynamics.The review cautions that this restricted setup does not support firm conclusions about ontological models in general.

12. Continuity

The review compares increasingly strong continuity requirements for ψ-epistemic models. Basic continuity is too weak for a no-go theorem, while stronger notions yield results but raise concerns about reasonableness and scope.

12. Continuity: Continuity requires nearby quantum states to have ontic representations with substantial overlap, linking Hilbert-space proximity to similarity in the ontic model.The basic definition imposes no specified relationship between the state-distance tolerance δ and overlap tolerance ϵ.
12. Continuity: Basic continuity cannot produce a no-go theorem because every pairwise ψ-epistemic model satisfies it.ABCL’s existence result therefore blocks this permissive continuity notion from ruling out ψ-epistemic models.
12. Continuity: PPM-δ-continuity requires a common ontic state with nonzero weight for sufficiently close collections of quantum states, not merely pairwise overlap.PPM prove that no such model reproduces the relevant quantum preclusions when δ ≥ 1 −√(d −1)/d.
12. Continuity: The review argues that PPM-δ-continuity is unreasonably strong because small weights in ontic distributions can make modest distribution changes appear discontinuous.A finite-state example reproduces the quantum predictions and is pairwise ψ-epistemic while satisfying only the basic continuity notion.
12. Continuity: Branciard showed that for Hilbert-space dimension d ≥4, Lipschitz continuity requires K=0, ruling out Lipschitz-continuous models.The review notes that Lipschitz continuity itself may be too strong because it imposes one fixed overlap-to-inner-product bound for all state pairs.
12. Continuity: Without Lipschitz continuity, existing overlap-ratio bounds concern states becoming almost distinguishable, leaving behavior for larger inner products open.Branciard’s family has R_n →0, but its minimum pairwise inner product also tends to zero.

13. Never ψ-ontic models

Never ψ-ontic models strengthen ψ-epistemic requirements by forbidding any uniquely identifying ontic region. Their existence across dimensions remains open, while weaker sometimes-ψ-ontology already has important consequences.

13. Never ψ-ontic models: A never ψ-ontic model requires every ontic region to be shared nontrivially by more than one quantum state.By contrast, a sometimes ψ-ontic model gives each pure state a region that no other pure state occupies.
13. Never ψ-ontic models: Sometimes ψ-ontology is sufficient to derive the implications associated with full ψ-ontology, including excess baggage and failure of maximal ψ-epistemicity.A sometimes ψ-ontic model must have at least as many ontic states as pure quantum states because each state receives its own region.
13. Never ψ-ontic models: The review proves that every sometimes ψ-ontic model is not maximally ψ-epistemic.A uniquely assigned region for one state contradicts the overlap condition required by maximal ψ-epistemicity.
13. Never ψ-ontic models: The Kochen–Specker model is never ψ-ontic, but existing ψ-epistemic models in dimensions at least three are sometimes ψ-ontic.Whether never ψ-ontic models exist in all dimensions is identified as an open question.

14. Discussion and conclusions

The review synthesizes ψ-ontology results, their assumptions, implications, criticisms, and remaining open questions. It emphasizes that ψ-epistemic models remain possible without auxiliary assumptions, while stronger exclusions remain unsettled.

14. Discussion and conclusions: The review organizes the subject into the ontic–epistemic distinction, prior arguments, no-go implications, three ψ-ontology theorems, and strengthened ψ-epistemic models.It aims to provide background, comprehensive criticism, and implications for researchers entering the area.
14. Discussion and conclusions: The Pusey–Barrett–Rudolph, Hardy, and Colbeck–Renner theorems all rely on auxiliary assumptions with varying degrees of reasonableness.The review specifically questions Hardy’s ontic-indifference assumption and Colbeck–Renner’s parameter-independence assumption.
14. Discussion and conclusions: Without auxiliary assumptions, ψ-epistemic models exist, motivating stronger notions of ψ-epistemicity to address phenomena such as state indistinguishability and no cloning.The review identifies bounds on overlaps between probability measures and quantum probabilities as a possible direction.
14. Discussion and conclusions: The most important open question is whether a never-ψ-ontic model exists for all pure states and projective measurements in dimension d ≥3.Other open problems concern deterministic conversions, symmetry-based exclusions, fixed-overlap families, POVMs, and quantum-information applications.
14. Discussion and conclusions: Experimental tests of quantum-state reality are not device independent because the quantum state is theory dependent and preparation devices must be assumed accurate.The review therefore calls for more theory-independent notions based on observed statistics.
14. Discussion and conclusions: Proving that the wavefunction is ontic would unify several no-go results, including Bell’s Theorem and exponentially growing ontic state spaces.The review presents this as a potential foundational and quantum-information-theoretic significance of ψ-ontology.

A. The ψ-ontic/epistemic distinction and objective chance

The appendix examines whether objective chance permits a distinction between ontic and epistemic states. Most objective-chance theories appear compatible with that distinction, but single-case propensity theories remain problematic.

A. The ψ-ontic/epistemic distinction and objective chance: The appendix asks whether theories involving objective chance support a distinction equivalent to the Bayesian ontic–epistemic distinction.The issue depends on whether chances are intrinsic properties of systems or depend on wider-world facts.
A. The ψ-ontic/epistemic distinction and objective chance: The ontic–epistemic status of quantum states depends on which probabilities are treated as objective chances, because ψ-ontology is defined through overlaps of state-representing measures.Measurement-device probabilities do not enter that definition.
A. The ψ-ontic/epistemic distinction and objective chance: Intrinsic objective chances require the relevant intrinsic properties of systems to determine equal chances across experimental contexts.The appendix contrasts intrinsic properties with descriptions involving ensembles or environmental conditions.
A. The ψ-ontic/epistemic distinction and objective chance: Most objective-chance theories admit an ontic–epistemic distinction because chances refer to nonintrinsic facts about systems.Single-case propensity theories are identified as the problematic exception.
A. The ψ-ontic/epistemic distinction and objective chance: Single-case propensity theory remains an unattractive basis for avoiding the question of whether quantum states are ontic or epistemic.The appendix notes that quantum theory itself motivates introducing objective chances.

B. The Kochen–Specker model

The Kochen–Specker model reproduces quantum predictions for the qubit while representing quantum states with overlapping probability measures. The appendix proves that it is maximally ψ-epistemic.

B. The ψ-ontic/epistemic distinction and objective chance: The Kochen–Specker model represents each quantum state with a unique probability measure over the Bloch sphere.The model’s density is used to calculate measurement probabilities.
B. The ψ-ontic/epistemic distinction and objective chance: The model is maximally ψ-epistemic because the overlap of its state-representing measures supplies the required quantum probability.The proof handles measure-one sets and uses absolute continuity on the relevant overlap region.
B. The ψ-ontic/epistemic distinction and objective chance: The model uses conditional probabilities for projective measurements defined through the Bloch-sphere representation and Heaviside step functions.The proof reduces integration to the region where the relevant step functions are positive.
B. The ψ-ontic/epistemic distinction and objective chance: The appendix proves that the model reproduces the quantum predictions for arbitrary pairs of qubit states and their orthogonal outcomes.The negative-angle case follows from the oddness of sine.
B. The ψ-ontic/epistemic distinction and objective chance: The maximal-epistemicity proof extends from a special support set to any measure-one set by showing that their difference has measure zero.This establishes the required overlap property for the full class of relevant sets.

C. Kochen–Specker contextuality

Kochen–Specker contextuality combines measurement noncontextuality with outcome determinism for projective measurements. The review relates this framework to ψ-ontology while distinguishing their implications.

C. Kochen–Specker contextuality: Operationally equivalent measurement implementations must receive the same conditional probabilities in a measurement-noncontextual model.Coarse-graining illustrates why different implementations of one POVM can otherwise require distinct conditional-probability distributions.
C. Kochen–Specker contextuality: KS noncontextuality requires outcome determinism and measurement noncontextuality for projective measurements.Outcome determinism makes projective-measurement outcomes certain at each ontic state, while measurement noncontextuality removes dependence on implementation context.
C. Kochen–Specker contextuality: KS noncontextual models cannot reproduce all projective measurements in Hilbert spaces of dimension ≥3.This is the Kochen–Specker no-go result under the combined requirements of measurement noncontextuality and outcome determinism.
C. Kochen–Specker contextuality: A model is KS contextual if it is measurement contextual or nondeterministic, since either property alone suffices to reproduce quantum predictions.The Beltrametti–Bugajski model exemplifies nondeterminism with measurement noncontextuality.
C. Kochen–Specker contextuality: ψ-ontology does not imply KS contextuality, because the two-dimensional Bell model is ψ-ontic but can be converted into a KS noncontextual model.KS contextuality also does not obviously imply the excess-baggage consequences associated with ψ-ontology.
C. Kochen–Speeker contextuality: Under the theorem’s conditions, KS noncontextuality forces the ontic-state support of each outcome to determine that outcome with certainty.Related results show that suitable overlap conditions yield a measure-zero revision that is KS noncontextual, including for maximally ψ-epistemic models.