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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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Properties5
StatusActive
Tagscontextuality, mathematics, quantum-foundations
SourceLIT-846 — Categories of Empirical Models, LIT-847 — A comonadic view of simulation and quantum resources
Extended 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
Cited as “grounds” byCLAIM-070 — Transport induced by one global stochastic kernel carries a global extension of the source model to a global extension of the target model, CLAIM-121 — A transport whose local kernels commute with restriction maps overlap-consistent source models to overlap-consistent target models, CLAIM-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, CLAIM-237 — The first open problem of directed transport is not yet posed: a contextual model is a menu of incompatible experiments, and under the natural order on menus any state-independent classical simulation is dominated automatically, so what is open is only the converse or a deficiency bound, CLAIM-245 — Under Contextuality-by-Default, a deterministic context-wise recoding can make a noncontextual signalling source maximally contextual, or turn a contextual source into pure signalling, so per-content signalling is not monotone under such maps and the quantity to bound is joint overlap discrepancy

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Source

Karvonen, Categories of Empirical Models (LIT-846), Definition 3.9 and Theorems 4.1, 4.3 and 4.5, as read in NOTE-651 (claims C1–C3, “strong (proof)”, proofs followed step by step). Abramsky, Barbosa, Karvonen and Mansfield, A comonadic view of simulation and quantum resources (LIT-847), Theorems 21 and 22, as read in NOTE-647 (claim C4, proof followed). Both are refereed (QPL 2018 in EPTCS, and LICS 2019).

What was actually shown

Take two finite measurement scenarios S and T, with possibly unrelated covers. A classical procedure from S to T answers each measurement x of T by measuring a set π(x) of S’s measurements and post-processing the outcomes. π sends every context of T into a context of S, so no incompatible set is attempted. Shared classical randomness is allowed: as a stochastic outcome map (LIT-846), or as an auxiliary non-contextual model, with adaptive choices (LIT-847). Each no-signalling empirical model d on S is pushed forward to one on T. When some procedure takes d to e, d simulates e.

  • (a) Non-contextuality is simulability from nothing. A model is non-contextual exactly when the model on the empty scenario simulates it (LIT-846 Theorem 4.1). A simulation from the empty scenario is a mixture of deterministic assignments, which is Fine’s form of a non-contextual model.
  • (b) The non-contextual fraction cannot fall. If d simulates e, then NCF(d) ≤ NCF(e) (LIT-846 Theorem 4.5; LIT-847 Theorem 21). The check: write d = λd₀ + (1 − λ)d₁ with d₀ non-contextual and λ = NCF(d). The pushforward F is affine, so F(d) = λF(d₀) + (1 − λ)F(d₁). F(d₀) is non-contextual, by (a) and composition of simulations. So NCF(e) ≥ λ. Equivalently, the contextual fraction cannot rise.
  • (c) Strong contextuality is reflected backwards. If e is strongly contextual and d simulates e, then d is strongly contextual (LIT-846 Theorem 4.3).
  • (d) No cloning. e simulates e ⊗ e exactly when e is non-contextual (LIT-847 Theorem 22). The proof first rules out NCF(e) = 0 by pigeonhole on the finitely many protocol components. It then rules out 0 < NCF(e) < 1, since (b) and NCF(e ⊗ e) = NCF(e)² would give NCF(e) ≤ NCF(e)².

What could have come out otherwise: a map that changes the cover and is built from locally classical pieces could have produced a global inconsistency the source lacked. It cannot, when the pieces form one procedure.

What this does not say

  • Which maps between model sets are simulations. That characterisation (LIT-845 Theorem 44, correct only in its v2 form) is THEORY-174’s, and is Proposed.
  • Nothing about signalling data. Every result assumes no-signalling. The pushforward is undefined without it (LIT-846 Remark 3.2), so Contextuality-by-Default systems are outside its reach.
  • Nothing about information preserved. The simulation preorder orders resources by contextuality. It is not shown to relate to Blackwell’s informativeness order (THEORY-156).
  • Not for LIT-846’s monoidal form of no-cloning. LIT-846’s Theorem 4.8 rests on a monoidal structure that the paper asserts without checking. (d) is cited to LIT-847 Theorem 22, whose proof was followed.

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