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LIT-845: Closing Bell: Boxing black box simulations in the resource theory of contextuality

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Properties6
StatusActive — read in full 2026-10-09 (NOTE-650); worth reading as the result that says which transformations between the empirical models of two different scenarios are classical. A probabilistic procedure S → T (a mixture of deterministic procedures, each answering every measurement of T by a context of S and a post-processing of outcomes) induces a convex map EMP(S) → EMP(T). A convex map is so induced exactly when it comes from a non-contextual model, satisfying a simpliciality predicate, on a new "hom" scenario [S, T] whose outcomes are procedures. Compatible local procedures that do not glue into one global procedure are contextual models of [S, T]: "contextual simulations". The construction makes scenarios with predicates a closed category. Non-adaptive only; the arXiv v2 corrects Theorem 44 of the published chapter.
Tagscontextuality, mathematics, quantum-foundations, logic
ExtendsLIT-847 — A comonadic view of simulation and quantum resources, LIT-846 — Categories of Empirical Models, LIT-016 — The sheaf-theoretic structure of non-locality and contextuality
SupportsCLAIM-125 — Directed transport between empirical models whose covers change has two open problems: when it preserves decision-relevant information, and how it extends to signalling data
Cited as “paper” byNOTE-650 — Closing Bell
Cited as “source” byTHEORY-1170 — A convex map between two scenarios' model sets is induced by a classical adaptive procedure exactly when some non-contextual model of the corresponding adaptive hom scenario induces it, THEORY-174 — Classical simulations between no-signalling empirical models on different scenarios never create contextuality, and a convex map between two scenarios' model sets is induced by a classical non-adaptive procedure exactly when some non-contextual model of the hom scenario that satisfies its predicate induces it

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Rui Soares Barbosa, Martti Karvonen and Shane Mansfield (2021), in A. Palmigiano and M. Sadrzadeh (eds), Samson Abramsky on Logic and Structure in Computer Science and Beyond, Outstanding Contributions to Logic 25, Springer (2023), pp. 475–529, DOI 10.1007/978-3-031-24117-8_13 — ARXIV-2104.11241

Key takeaways

  • Procedures between scenarios. A deterministic procedure S → T is a simplicial relation π from T’s measurements to sets of S’s measurements, plus outcome maps α_x : E_S(π(x)) → O_T,x (Definition 17); probabilistic procedures are convex mixtures of these (Definition 21), so π may be random too, which is more general than Karvonen’s (Remark 28). A procedure f induces EMP(f) : EMP(S) → EMP(T), which preserves convex combinations (Lemma 24).
  • Contextuality as non-simulability (Theorem 29): e is (probabilistically / logically / strongly) contextual iff no (probabilistic / possibilistic / weak) simulation from the empty model reaches it. Non-local games and Bell functionals are probabilistic procedures into the one-bit scenario [2] (§3.5).
  • Which maps are classical. A convex map is fixed by its values on deterministic models (Theorem 37, via signed global sections). A deterministic-preserving map is induced by a deterministic procedure iff each context of T is computed from one context of S (Theorem 40). In general (Theorem 44, v2): F is induced by a probabilistic procedure iff F = F_e for some non-contextual model e of the hom scenario [S, T] satisfying the predicate g_{S,T} (each target context uses only a context of S). Deciding this is a linear program over a projection of the non-contextual polytope.
  • Contextual simulations. A model of [S, T] is a family of compatible local procedures, one per context of T; it glues to one global procedure exactly when it is non-contextual (§4.5). Contextual models of [S, T] are named “contextual simulations” and left open (§6.7).
  • Closure. Scenarios equipped with structure predicates, with procedures preserving them, form a closed category with [−, −] as internal hom (Theorem 46, Corollary 47).
  • What changed in v2. v1 encoded each convex map F as one canonical model e_F, on the claim that F’s decomposition into deterministic experiments is unique, and stated Theorem 44 as “e_F is non-contextual”. v2 drops the canonical encoding and asks only for some non-contextual e with F = F_e. The published chapter carries the v1 statement.

Standing in the record

Filed on 2026-10-09 at the owner’s direct request, as the third of the works on simulations between empirical models. It is not from the manuscript bibliography: the owner asked for it directly on 2026-10-09. It was registered with LIT-846 and LIT-847 because it directly continues their framework: Remark 28 revises both their morphisms, and Theorem 44 answers the question their categories raise, which maps between models are free. It is a source of THEORY-174. Its prologue names natural language among the settings where the local–global tension recurs, citing Wang et al. (LIT-842). NOTE-650 says how it bears on the manuscript’s transport problem on line pragmatic-transport.

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