REVIEW 2 major objections 4 minor 13 references
A No-Go Theorem for {\psi}-ontic Models? Yes! Response to Criticisms
T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The paper defends its no-go theorem against Gao's criticism: Harrigan-Spekkens psi-ontic models already assume classical probability mixtures, so the entropy mismatch is a theorem, not an extra premise.
desk verdict A clear, honest reply that correctly pins the classical-mixing premise on HS, but it does not close the gap Gao pointed to: the argument shows Shannon entropy fails, not that no entropy on ontic states can work. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the epistemic state of the Harrigan-Spekkens framework: a probability measure $p(\lambda|P)$ over the ontic state space $\Lambda$, with a mixture of pure states represented as $\sum_i p_i\, p(\lambda|P_{\psi_i})$. The argument's engine is the entropy conflict: because this is a Kolmogorovian probability measure, the Shannon/Gibbs entropy is the natural and only thermodynamic entropy on the space of such measures, but the quantum mixture it represents requires the von Neumann entropy, which cannot be computed from a classical measure. The $\sigma$-algebra structure of a Kolmogorov space is what makes all ontic states distinguishable in principle.
What would settle it
Render the theorem false by constructing an explicit Harrigan-Spekkens psi-ontic model whose epistemic states are classical probability measures over ontic states and that still reproduces the von Neumann entropy of, say, an equal mixture of spin-up along z and spin-up along x (or that derives the Holevo bound). Producing such a model with its density-matrix outcome probabilities would directly contradict the no-go claim.
Extended reading notes
Core claim
Gao's objection was that the entropy argument implicitly assumes non-orthogonal ontic states are classically distinguishable. The authors reply that no extra assumption was made: a classical probability measure over ontic states automatically distinguishes every element of its sigma-algebra, and the Harrigan-Spekkens framework chooses such measures for mixtures. Therefore the mismatch between Shannon/Gibbs entropy and von Neumann entropy follows from the framework's own premises. Saying 'use the von Neumann entropy' is, in their view, an admission that the model cannot compute the entropy from the epistemic state, confirming the no-go result.
Load-bearing premise
The whole result collapses if the Harrigan-Spekkens framework is not actually committed to representing quantum mixtures as classical probability distributions over ontic states; the paper insists the commitment is there, but that is exactly what a defender of HS would have to deny.
Editorial extensions
If this is right
- A Harrigan-Spekkens psi-ontic model cannot reproduce quantum statistical mechanics, quantum thermodynamics, or quantum information theory (e.g., the Boltzmann distribution or the Holevo bound), so it cannot be a full model of quantum mechanics.
- The PBR theorem, because it presupposes the Harrigan-Spekkens framework, is not evidence for a realist reading of the quantum state; it only rules out HS psi-epistemic models.
- The HS framework is internally too classical: its classical mixture rule already distinguishes all ontic states, and adding more structure to hide this at the level of measurements does not fix the mixed-state entropy problem.
- The no-go theorem does not rule out all realist interpretations; models like Bohmian mechanics may not satisfy the HS operational definitions, so their status is left open.
Reading between the lines
- If the theorem holds, the Harrigan-Spekkens classification scheme turns out to be a poor measuring stick for the ontic/epistemic debate: both branches of the dichotomy fail, and the dichotomy itself is the problem.
- The entropy test suggests a practical diagnostic for any proposed ontological model: check whether statistical mixtures are classical measures; if they are, a Shannon-von Neumann mismatch is likely to appear in some regime.
- The argument could be pressed further: the same classical-mixture assumption may undermine not just HS models but any hidden-variable program that represents quantum mixtures as classical ensembles, which is a broader target than the paper explicitly claims.
- One concrete quantity to study is the gap between Shannon and von Neumann entropy for a two-state mixture; that gap quantifies exactly how much classical distinguishability the HS model wrongly adds.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This note responds to Shan Gao's criticism of the authors' earlier no-go theorem for ψ-ontic models in the Harrigan-Spekkens (HS) framework. The authors argue that the allegedly implicit assumption of classical mixing of epistemic states is in fact already part of the HS framework, so the use of Shannon/Gibbs entropy is justified. They further claim that Gao neither addresses the outstanding problems of recovering quantum statistical mechanics, thermodynamics, and information theory from an ontological model, nor explains how to compute a von Neumann entropy from an epistemic state. The paper concludes that Gao's criticism is rejected, reiterates that the no-go theorem is correct, and draws broader consequences for the interpretation of the PBR theorem.
Significance. If fully established, the no-go theorem would be a significant result in quantum foundations: it would show that ψ-ontic models as defined by Harrigan and Spekkens cannot reproduce quantum mechanics, thereby undermining a widely used classification scheme and altering the standard reading of the PBR theorem. The present response does clarify the structure of the authors' argument and honestly lists several open problems, which is a strength. However, the central logical gap identified by Gao and by the stress-test review remains: the step from the HS mixing rule to the unique applicability of Shannon/Gibbs entropy is not proved. The paper's conclusion therefore overstates what its premises establish, and the response functions more as a clarification of the authors' position than as a definitive refutation of the criticism.
major comments (2)
- [Section 2, premise 4] The inference from 'mixtures in the HS model are modeled as classical probability distributions' to 'we should be entitled to use the only tools we have that work on those (i.e. classical statistical mechanics)' is not a mathematical consequence. A Kolmogorov probability measure on the ontic space admits many entropy functionals, such as Rényi entropies or other measures satisfying different axioms. The paper itself concedes in Section 3 that 'one may give a different definition of entropy to use on the ontic space' and that 'we are open to the possibility that there may be something that can be made to work.' This concession directly undermines the no-go conclusion: the argument at most demonstrates a conflict between the HS convex-mixing rule plus Shannon/Gibbs entropy and quantum thermodynamics, not the impossibility of all ψ-ontic HS models. To sustain the theorem, the authors must either prove that the HS definitions force the Shannon/Gibbs entropy, or rule out all alternative entropy functionals, neither of which appears in this response.
- [Section 2, §2 response to Gao] The central defense—that the classical-mixing assumption is 'an assumption made by Harrigan and Spekkens'—rests on a reading of HS that is exactly what Gao contests. The quotes from Gao and HS establish that the HS framework represents mixtures by weighted sums of probability measures, but they do not show that this entails Shannon/Gibbs entropy as the entropy functional for epistemic states. The paper relies on the authors' prior article (Carcassi et al. 2024) for this interpretation, yet that article is precisely the subject of the dispute, so the argument is circular. To reject Gao's criticism, the response must provide an independent derivation from the HS definitions that the Shannon/Gibbs entropy is the unique or forced choice, or explicitly acknowledge that the no-go theorem is conditional on this choice.
minor comments (4)
- [General] The response is not self-contained: the no-go theorem is not stated in this note, and the reader is referred to Carcassi et al. (2024). Stating the theorem and a brief proof sketch would make the paper more accessible and avoid ambiguity about the exact claims being defended.
- [Section 3] The phrase 'raised by Carcassi in a signed response as part of the peer-review process' refers to an unpublished process and is not verifiable; it should be removed or replaced with a citation to a publicly available document.
- [Section 4] The discussion of Bohmian mechanics, while relevant to the scope of the no-go theorem, is only loosely connected to the direct response to Gao and could be condensed or moved to a separate paper to keep the reply focused.
- [Various] Occasional rhetorical overstatement, such as 'the theorem proved in our paper is correct, and therefore true' and 'the HS model framework is fundamentally flawed,' conflicts with the paper's own acknowledgment that alternative entropy definitions might work; more careful phrasing would better match the actual strength of the argument.
Circularity Check
The no-go conclusion is conditional on a chosen entropy functional: the paper assumes Shannon/Gibbs entropy on epistemic states, then infers failure when it does not match von Neumann entropy, while conceding alternative entropy definitions remain open.
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ansatz smuggled in via citation
[Section 2, schematic premises 4–5 and quoted passage from Carcassi et al. (2024), p. 5; Section 3, concession paragraph.]
"Given that M(Λ) is the space of probability measures over Λ, it is natural to use the Shannon/Gibbs entropy function HΛ : M(Λ) → R, as this is what is typically done in statistical mechanics ( ibid., p. 5). ... We are open to the possibility that there may be something that can be made to work, though we are skeptical that this can be done with the simple 'quantum state at time t ↔ ontic state at time t' relationship assumed by the HS categorization."
The paper's load-bearing premise is that the entropy of an HS epistemic state must be the Shannon/Gibbs entropy because epistemic states are probability measures. That identification is not derived from Harrigan and Spekkens' definitions; it is an entropy functional chosen in the authors' prior work and defended here only by an appeal to 'the only tools we have.' The no-go conclusion therefore reduces to the tautology that Shannon/Gibbs entropy on a Kolmogorov measure need not equal von Neumann entropy. The paper itself concedes that a different entropy on the ontic space might work, so the argument proves a conditional conflict between the HS mixing rule plus a specific entropy choice and quantum thermodynamics, not a theorem against all ψ-ontic HS models.
full rationale
The paper is not wholly circular: its central premise that HS models represent mixtures by convex combinations of probability measures is grounded in Harrigan and Spekkens' framework as quoted through Gao, and the claim that Kolmogorovian probability cannot reproduce all quantum statistics is independently known. However, the decisive step from 'epistemic states are probability measures' to 'therefore the relevant entropy is Shannon/Gibbs' is an assumption imported from the authors' earlier paper, not a consequence of HS definitions. The authors explicitly allow alternative entropy definitions, which means their theorem does not rule out all ψ-ontic HS models unless such alternatives are impossible, and the response does not prove that impossibility. Thus the central no-go result is partially sustained by a self-cited entropic ansatz, though it retains independent content in its critique of classical mixing. Score 4 reflects this partial, not total, circularity.
Assumptions & free parameters
assumptions (5)
- domain assumption The HS ontological model treats a quantum statistical mixture as a classical probability distribution over ontic states, so all ontic states are elements of a Kolmogorov probability space.
- domain assumption A valid ontological model must reproduce all results of quantum mechanics, including quantum statistical mechanics, thermodynamics, and quantum information theory.
- domain assumption The Shannon/Gibbs entropy is the entropy functional appropriate to classical probability measures, and the von Neumann entropy is not directly computable on an epistemic state.
- standard math A Kolmogorov probability space permits exhaustive distinguishability of ontic states via indicator random variables, which is what makes the HS framework too classical.
- domain assumption If the HS framework is fundamentally flawed, then the PBR theorem, which presupposes that framework, cannot support metaphysical conclusions about the quantum state.
Cite this review
Pith. "Pith review of A No-Go Theorem for {\psi}-ontic Models? Yes! Response to Criticisms." pith.science (2026). https://pith.science/paper/CTJW4MAQ
@misc{pith2026241219182,
author = {Pith},
title = {Pith review of: A No-Go Theorem for \psi-ontic Models? Yes! Response to Criticisms},
year = {2026},
howpublished = {\url{https://pith.science/paper/CTJW4MAQ}},
note = {Machine review of arXiv:2412.19182}
}
read the original abstract
This short note addresses the criticisms recently proposed by Shan Gao against our article "On the Reality of the Quantum State Once Again: A No-Go Theorem for {\psi}-Ontic Models" (Found. Phys. 54:14). The essay aims to respond to such objections and to show once again that the theorem proved in our paper is correct, and therefore true - contrary to Gao's claims. Philosophical consequences of this fact are briefly discussed.
Reference graph
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Carcassi, G., Oldofredi, A., and Aidala, C. A. (2024). On the reality of the quantum state once again: A no-go theorem for ψ -ontic models. Foundations of Physics , 54:14
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Harrigan, N. and Spekkens, R. (2010). Einstein, Incomplete ness, and the Epistemic View of Quantum States. Foundations of Physics , 40:125–157
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Reviewed August 11, 2026 · model on record in the stance chip above.
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