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Classical explainability does not follow from local classicality: bilocal classical theory is explainable, yet some locally-classical theories are not.

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2026-08-03 20:32 UTC pith:NMWDXEQW

load-bearing objection The BCT ontological model is a genuine achievement; the no-go half of the decoupling claim currently rests on an unproved equation, so the paper is not fully there yet. the 2 major comments →

arxiv 2511.19266 v2 pith:NMWDXEQW submitted 2025-11-24 quant-ph

Decoupling local classicality from classical explainability: A noncontextual model for bilocal classical theory and a locally-classical but contextual theory

Sina Soltani , Marco Erba , David Schmid , John H.Selby This is my paper
classification quant-ph
keywords operational probabilistic theoriesgeneralised noncontextualityontological modelsbilocal classical theorylatent classical theorieslocal tomographyclassical explainability
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper decouples two notions of classicality: a theory being locally classical (every system is a simplex theory) and being classically explainable (admitting an ontological model into standard classical theory). It proves that bilocal classical theory (BCT), a locally-classical theory that violates local tomography, admits an ontological model, refuting a conjecture that it might not. It then constructs locally-classical theories (latent classical theories, or LCTs) with non-full-rank latent states and proves they admit no ontological model. Together, these results show that local classicality neither implies nor prevents classical explainability, and that the compositional structure of a theory, not just its local state spaces, determines whether it is classically explainable.

Core claim

The paper establishes two theorems. Theorem 2.2: there exists an ontological model for bilocal classical theory, a strongly causal operational theory whose composite systems are classical simplices but whose composition rule doubles bipartite dimensions, so composite states are not determined by local measurements. The model maps every system of dimension n to a classical system of dimension 2n, encoding a binary flag that tracks the non-local 'phase' degrees of freedom through controlled-NOT and copy operations, and is shown to be linear, diagram-preserving, probability-preserving, and determinacy-preserving. Theorem 3.2: no ontological model exists for any latent classical theory in which

What carries the argument

For BCT, the central object is the explicit ontological map: each system n is sent to a 2n-dimensional classical system, pure states acquire an extra classical bit (the 'binary phase'), and atomic transformations are implemented with controlled-NOT and copy gates. This is the first full-theory ontological model for a non-locally-tomographic theory. For the no-go result, the load-bearing objects are the latent-theory construction applied to classical systems—specifically, the existence of a non-null bipartite effect that annihilates every product state (Eq. 88)—together with the finite-dimensional Choi-swap identity of classical theory (Eq. 91), which together force the contradiction.

Load-bearing premise

The no-go theorem leans on the latent-theory construction of Ref. [75]—assumed without proof here to yield valid operational theories in which a rank-deficient latent state produces a non-null bipartite effect annihilating every product state (Eq. 88)—and on the finite-dimensional Choi-swap identity in classical theory (Eq. 91) as an unproved standard lemma; if either premise fails, Theorem 3.2 does not establish non-explainability.

What would settle it

Verify directly within the construction of Ref. [75] applied to classical systems that a rank-deficient latent state yields a non-null bipartite effect b satisfying (κ⊥|κ)=0 and hence Eq. (88), and check that the Choi-swap identity of Eq. (91) holds in the classical theory used as the model's image. If either fails—the 'annihilating' effect turns out to be null, or the Choi/swap relation is not valid for the proposed ontic space—Theorem 3.2's contradiction dissolves.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Bilocal classical theory is classically explainable, refuting the conjecture from its original presentation and showing that violating local tomography does not by itself prevent a theory from having a noncontextual ontological model.
  • Latent classical theories with a non-full-rank latent state are locally classical yet admit no ontological model, so local classicality and classical explainability are logically independent.
  • The assumption of local tomography in the structure theorem of Ref. [2] is a genuine limitation: BCT falls outside that theorem's scope precisely because it violates local tomography, yet it is still classically explainable.
  • The BCT model is, to the paper's knowledge, only the second ontological model for an entire theory and the first for a non-locally-tomographic theory, providing a concrete methodology for constructing models of other theories with modified composition rules.
  • The contradiction in Theorem 3.2 identifies the existence of a non-null bipartite effect annihilating all product states as the operative obstruction to classical explainability in locally-classical theories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These results suggest a classification principle: a locally-classical theory may be classically explainable exactly when no non-null bipartite effect annihilates all product states; other foil theories with modified composition rules (such as the twirled and swirled worlds mentioned in the paper) could be tested against this criterion.
  • The extra binary flag in the BCT model functions as a hidden phase variable, hinting that locally-classical theories obtained by 'phase-space doubling'—where each subsystem carries a discrete gauge bit—will generally admit ontological models, pointing toward systematic constructions for other exotic composition rules.
  • The no-go theorem implies that any locally-classical theory with a non-full-rank latent state will violate generalized noncontextuality inequalities in some bipartite scenario, even though its unipartite sectors are simplex-embeddable; this provides a concrete, testable distinction between BCT and LCTs.
  • If the forthcoming generalization of the proof to infinite-dimensional classical embeddings succeeds, the obstruction will be purely structural and not an artifact of finite-dimensional ontic spaces, making the independence of local classicality and explainability a robust feature of any classical ontology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper defines operational probabilistic theories (OPTs) and the notion of an ontological model as a linear, diagram-preserving, probability-preserving map into classical theory. Its main positive result, Theorem 2.2, is an explicit ontological model for bilocal classical theory (BCT), a locally-classical theory violating local tomography; this is claimed to refute a conjecture from Ref. [1] and to be the first ontological model for a non-locally-tomographic theory. The main negative result, Theorem 3.2, claims that latent classical theories with at least one non-full-rank latent state admit no ontological model. Together these are taken to show that local classicality and classical explainability are logically independent. The positive construction is supported by lengthy appendices checking linearity, diagram preservation, probability preservation, and determinacy. The negative theorem, however, relies on a non-derived identity, Eq. (89), and on an external, non-peer-reviewed construction [75] whose restriction to classical sectors is not proved in the paper.

Significance. If both theorems are correct, the paper settles a question of real interest in the OPT literature: it gives a fully compositional ontological model for a theory that fails local tomography, and it exhibits a locally-classical theory that is nevertheless not classically explainable. The positive construction is substantial and largely self-contained, with explicit diagrammatic checks; this is a genuine strength. The decoupling conclusion, however, is only as strong as Theorem 3.2, and that theorem currently rests on unverified external input and an omitted proof. The paper is honest enough to flag these dependencies, but they are load-bearing. The result would be important and publishable after the missing computations and the validity of the latent classical theory construction are supplied in the manuscript itself.

major comments (2)
  1. [§3.2, Eq. (89)] The proof of Theorem 3.2 hinges on the identity that for every bipartite state β, the composition of β with the effect (b| and a partial trace yields the null process. This is stated as 'easily verified by direct computation resorting to Ref. [75, App. B]', but no computation is shown. It does not follow from Eq. (88), which only concerns product states; indeed, the whole point of these theories is that local tomography fails, so product states do not span all bipartite states. The contradiction in Eq. (92) uses Eq. (89) essentially. Without a full derivation, the no-go theorem is incomplete. Please provide the direct computation as a lemma in the paper or an appendix.
  2. [Definition 3.1; Eqs. (87)-(90)] Latent classical theories are defined by applying the construction of Ref. [75] to classical systems, or as the 'restriction' of LQTs to their classical/diagonal sectors. No proof is given that this restriction yields a valid OPT satisfying the axioms used in Theorem 3.2: the composition rule Eq. (87), the existence of the effect (κ⊥|, the validity of the composite effect (b| defined by a tensor product of an effect on L12 with discarding effects on C1 and C2, and the identity Eq. (89). Since Ref. [75] is an unpublished preprint and the restriction step is not formalized, the counterexample family is not self-contained. The manuscript should include a self-contained construction of the LCTs used, or at least a precise lemma establishing their OPT status and the properties (88)-(90).
minor comments (5)
  1. [Definition 1.2] The sentence 'every scalar in Φ to a scalar in Θ' appears to be backwards: a map from Θ to Φ should send scalars of Θ to scalars of Φ. Please correct.
  2. [§2, Eq. (77)] The image of a single-system pure state is drawn as 'i n B' without specifying the binary label s used in Eq. (71) for states of dimension 2n. Please state explicitly that the binary component is fixed (e.g., s=0) or otherwise clarify the notation.
  3. [§3.2, Eq. (91)] The generalized Choi identity is invoked as 'we recall' with no proof. It is standard for finite-dimensional locally tomographic theories, but since it is used in the no-go contradiction, it should be stated and proved as a lemma, or a precise reference should be given.
  4. [App. A.2.1] The discussion of diagram preservation for µ is terse and at points reads as circular ('must yield the same result as the rightmost diagram'). The equality can be verified directly from Eqs. (78), (112), and (118); please rewrite this subsection as an explicit computation rather than an assertion.
  5. [Throughout] There are numerous typos and minor grammatical errors, e.g., 'reults', 'every every state', 'digram', 'an arbitrary transformationst', and an apparently misplaced phrase in Definition 1.2. A careful proofread is needed.

Circularity Check

0 steps flagged

No significant circularity: the two theorems are supported by explicit constructions and a contradiction argument, not by redefining inputs as outputs.

full rationale

Theorem 2.2 is a constructive existence result: the ontological map in Definition 2.1 is specified explicitly on pure states, effects, and atomic transformations, and the consistency checks in Section 2.1 and Appendix C are direct diagrammatic computations within classical theory. The only imported formal ingredient is the transformation-level simplification of ontological models from the authors' companion work [6]; that result does not presuppose the existence of a BCT model, so it is framework-level rather than a circular premise. Theorem 3.2 is a proof by contradiction: assuming an ontological model exists, the LCT relation (89) is mapped under diagram preservation/linearity to a null classical process, which is then contradicted by the non-null probability of (90). The LCT construction is imported from Ref. [75] (with overlapping authorship), and Eq. (89) is asserted rather than fully derived in the present paper; these are verification and correctness gaps, not circular reductions. Equation (89) is not the target non-existence statement, and the non-existence conclusion is not used to derive it. The Choi identity (91) is a standard finite-dimensional locally-tomographic fact. There are no fitted parameters, no empirical predictions, and no step in which a claimed result is defined as its own input. The self-citations are load-bearing only at the level of definitions/construction and do not make the central derivations equivalent to their assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No numerical free parameters: the paper is purely structural. No new physical degrees of freedom are postulated: the latent factors come from the prior latent-theory construction [75], and the ontic spaces in the BCT model are ordinary finite-dimensional classical systems.

axioms (5)
  • domain assumption BCT is a strongly causal OPT with composition rule D(n,m)=2nm for n,m != 1, bilocal tomography, and the pure-state/effect rules of Eqs. (35)-(44).
    Imported from Ref. [1]; the existence proof takes BCT exactly as defined there and does not reprove its consistency.
  • domain assumption The definition and properties of ontological models for strongly causal OPTs—linear, diagram-preserving, probability-preserving maps to classical theory—are as in Ref. [6]; for strongly causal OPTs, instrument structure can be ignored.
    This formalizes the target concept; if this framework is not the right notion of classical explainability, Theorem 2.2 loses its advertised interpretation.
  • domain assumption Latent classical theories, as defined by applying the construction of Refs. [75, Apps. A-B] to classical systems, are valid OPTs with latent factors L_12 and latent states kappa; the non-full-rank case yields a non-null bipartite effect annihilating all product states (Eq. 88).
    The no-go theorem 3.2 is conditional on this construction; the detailed consistency proof is deferred to [75].
  • standard math Finite-dimensional classical theory contains generalized Choi vectors and covectors satisfying the swap identity (Eq. 91).
    Used in the contradiction argument of Theorem 3.2; standard in Choi/process-matrix representations but not proved in the paper.
  • domain assumption Every system in the theories considered has a unique decomposition into elementary systems.
    Assumed so that parallel composition can be specified by elementary systems, following Refs. [5] and [75].

pith-pipeline@v1.3.0-alltime-deepseek · 33468 in / 19633 out tokens · 215125 ms · 2026-08-03T20:32:51.953938+00:00 · methodology

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read the original abstract

We construct an ontological model for the theory known as bilocal classical theory doi.org/10.1103/PhysRevA.102.052216. To our knowledge, this is only the second time that an ontological model has been constructed for an entire theory, rather than just for some particular scenarios within a theory. This result refutes a conjecture from doi.org/10.1103/PhysRevA.102.052216 which suggested that there might be no local-realist ontological model for bilocal classical theory. Moreover, it is the first time that an ontological model has been constructed for a theory that fails to be locally tomographic, showing that the assumption of local tomography underpinning the structure theorem in doi.org/10.22331/q-2024-03-14-1283 is a genuine limitation of the theorem. This demonstrates that in general there is no tension between failures of local tomography and classical explainability (i.e., generalised noncontextuality). In fact, bilocal classical theory is in many ways more simply understood via the underlying ontological model than it is within its original formulation (much as how odd-dimensional stabiliser subtheories can be more simply understood via Spekkens' toy theory). Furthermore, this result naturally leads to the question, does every locally-classical theory admit of an ontological model? By constructing a concrete counterexample, we show that this is not the case. Our findings demonstrate that there is no straightforward relationship between theories being locally-classical, and them being classically-explainable. This shows that the fundamental status of compositional properties (such as local tomography) is not a technical side-issue, but a central and unavoidable question for a coherent understanding even of classicality itself.

Figures

Figures reproduced from arXiv: 2511.19266 by David Schmid, John H.Selby, Marco Erba, Sina Soltani.

Figure 1
Figure 1. Figure 1: Venn diagram incorporating our results. One example of a theory is included for each region, with the [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗

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