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LIT-847: A comonadic view of simulation and quantum resources

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Properties7
StatusActive — read in full 2026-10-09 (NOTE-647); worth reading as the paper that makes simulation between empirical models adaptive and shows it is the same thing as convertibility by free operations. Measurement protocols (decision trees of compatible measurements) form a comonad MP on scenarios and on empirical models; a simulation of e by d is a deterministic map MP(d ⊗ c) → e with c non-contextual. Such a simulation exists exactly when e is obtained from d by a term in the free operations (relabelling and translation of measurements, coarse-graining, mixing, choice, tensor, and the new conditional measurement), with an equational theory and normal forms. Along any simulation the non-contextual fraction cannot fall, and no contextual model can be cloned.
Tagscontextuality, mathematics, quantum-foundations
ExtendsLIT-846 — Categories of Empirical Models, LIT-265 — The contextual fraction as a measure of contextuality, LIT-016 — The sheaf-theoretic structure of non-locality and contextuality
Extended byLIT-845 — Closing Bell: Boxing black box simulations in the resource theory of 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-647 — A comonadic view of simulation and quantum resources
Cited as “source” byTHEORY-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, THEORY-720 — Classical simulations between no-signalling empirical models on finite scenarios, which may change the cover, never create contextuality: a model is non-contextual exactly when the empty model simulates it, the non-contextual fraction cannot fall along a simulation, strong contextuality is reflected backwards, and only non-contextual models can be cloned

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Samson Abramsky, Rui Soares Barbosa, Martti Karvonen and Shane Mansfield (2019), in 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pp. 1–12, DOI 10.1109/LICS.2019.8785677 — ARXIV-1904.10035

Key takeaways

  • Free operations, with one new. The operations that build one empirical model from others without contextual resources (§III-A): the zero and singleton models, translation of measurements along a simplicial map (which may change the cover), coarse-graining of outcomes, probabilistic mixing, controlled choice (&), tensor product (⊗), and the new conditional measurement x?y (measure x, then choose the next measurement from the outcome). CF is monotone for each (Proposition 7); closed terms are exactly the non-contextual models (Proposition 8).
  • An equational theory with normal forms. Equations (1)–(28) are sound up to isomorphism of models (Proposition 10); every term rewrites to a mixture of terms of the form (f* (base ⊗ … )[x?y]…)/h (Proposition 11). Completeness is open.
  • Measurement protocols as a comonad. Protocols are prefix-closed sets of runs (Definition 14); MP(X) is the scenario of all protocols over X, MP(e) the induced model, and MP is a comonoidal comonad on scenarios and on empirical models (Theorem 17). A simulation of e by d is a deterministic morphism MP(d ⊗ c) → e for some non-contextual c (Definition 18): adaptive, with shared classical randomness.
  • Simulation equals free convertibility (Theorem 20): d simulates e iff e ≅ t[d/v] for a typed term t in the free operations. Hence NCF(d) ≤ NCF(e) (Theorem 21) and no-cloning: e simulates e ⊗ e iff e is non-contextual (Theorem 22).
  • Examples drawn from the literature. Barrett and Pironio’s simulation of any two-output bipartite box by PR boxes, and their five-partite quantum box that no number of PR boxes simulates, are instances (Example 19).

Standing in the record

Filed on 2026-10-09 at the owner’s direct request, as the second of the works on simulations between empirical models, following Karvonen’s Categories of Empirical Models (LIT-846). It is not from the manuscript bibliography: the owner asked for it directly on 2026-10-09. It is a source of THEORY-174. NOTE-647 says how it bears on the manuscript’s transport problem on line pragmatic-transport.

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