Contribution
Before this paper, morphisms between empirical models (Mansfield’s thesis, Wester) were deterministic and sent each measurement to one measurement. This paper allows a target measurement to be answered by a set of source measurements and allows the outcome map to be stochastic, using the Kleisli category of the distribution monad. With these morphisms it recasts known facts about contextuality as facts about one category, Emp_R, and proves a no-cloning theorem for contextual models. The category is offered as a resource theory of contextuality.
Key insight
One empirical model can stand in for another when a team of non-communicating agents, each told only which target measurement to perform, can answer it by measuring a fixed set of the source model’s measurements and processing the outcomes with shared classical randomness. Classical processing of this kind cannot manufacture a global inconsistency that was not already in the source. So contextuality is what cannot be simulated from nothing, and every simulation is monotone for it.
Assumptions
- Finite scenarios. ⟨X, M, (O_x)⟩ with X finite, each O_x finite, and M an antichain covering X (Definition 2.1).
- Distributions over a semifield R (R≥0, ℝ or the Booleans), so that conditional distributions exist (Definition 2.3). Only Theorem 4.7 uses the semifield restriction.
- No-signalling built in. Empirical models are compatible families (Definition 2.6). Remark 3.2: the pushforward is well defined only because d restricted to π(C) does not depend on which source context contains π(C), so signalling models are excluded by necessity, not by choice.
- Morphisms point the other way from the map of complexes. A morphism Y → X has π : X → Y (footnote 1), chosen so that maps from the terminal object are global sections.
- No preprocessing. π is fixed: no randomised or adaptive choice of which source measurements to perform (§5).
Key results
- Definition 3.1 (deterministic morphism). A simplicial relation π : X → Y (every target context C has π(C) inside some source context) and a natural transformation σ : E_Y(π(−)) → E_X(−). Pushforward: (σ_*d)_C = D_R(σ_C)(d|π(C)). Lemma 3.3: since E_X is a sheaf, such σ glue uniquely along any cover.
- Definition 3.9 (R-stochastic morphism, simulation). As above with σ : E_Y(π(−)) → D_R ∘ E_X(−); composition is Kleisli. A simulation d → e is a morphism with σ_*d = e. Lemma 3.12: stochastic σ glue along partitions (by independent product), not uniquely and not along arbitrary covers, because D_R ∘ E_X is not a sheaf.
- Proposition 3.13. Simulations of e_1, …, e_n from d with the same π mix into a simulation of their convex combination.
- Theorem 4.1. Simulations 1 → e from the terminal model on the empty scenario correspond bijectively to global distributions explaining e; so e is R-non-contextual iff a simulation 1 → e exists.
- Theorem 4.2. A semifield homomorphism R → S gives a functor Emp_R → Emp_S preserving the terminal object; so S-contextuality of the image implies R-contextuality, and logical contextuality implies probabilistic contextuality. The possibilistic-collapse functor is neither surjective on objects nor full.
- Theorem 4.3. For R = R≥0 or the Booleans, if d → e and e is strongly contextual, then d is strongly contextual.
- Lemma 4.4 and Theorem 4.5. Pushforward preserves convex mixtures, so a decomposition d = λd^NC + (1 − λ)d′ pushes to one of e; NCF(d) ≤ NCF(e) whenever d → e. Equivalently CF(e) ≤ CF(d).
- Theorem 4.7. If x lies in exactly one maximal context S, there is a simulation e|X{x} → e (sample x from the conditional e_S(− | s)). So any model on a cover reducible to nothing by Graham reductions is non-contextual: Vorob’ev’s theorem in this language.
- Theorem 4.8 (no-cloning). With e ⊗ d on the join of covers, a simulation e → e ⊗ e exists iff e is non-contextual. The proof is by induction on |X|, using that π from the n-fold copy back to X must either cover X in every copy (forcing X itself to be a context) or miss part of it in some copy.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | A model is non-contextual iff it can be simulated from the empty model | strong (proof) | Theorem 4.1 |
| C2 | Simulation never increases the contextual fraction, across scenarios with different covers | strong (proof) | Lemma 4.4, Theorem 4.5 |
| C3 | Simulation reflects strong contextuality | strong (proof) | Theorem 4.3 |
| C4 | Acyclic covers admit only non-contextual models, via simulations | strong (proof) | Theorem 4.7 |
| C5 | A model can be cloned by classical simulation iff it is non-contextual | strong (proof), monoidal structure asserted | Theorem 4.8 |
| C6 | The morphisms capture “simulation with shared classical randomness” fully | not claimed; open | §5: preprocessing is missing |
Method
Category theory over the sheaf-theoretic framework. Scenarios are simplicial complexes with an outcome sheaf; morphisms are pairs of a simplicial relation and a (Kleisli) natural transformation, by analogy with morphisms of ringed spaces. Results are proved by turning a property of a model into the existence of an arrow (from the terminal object, or to a product with itself).
Concepts
- simplicial relation: a relation π : X → Y such that the image of every face is contained in a face.
- event sheaf E_X: U ↦ ∏_{x∈U} O_x, with projections.
- simulation d → e: a morphism of scenarios whose pushforward takes d to e; “d simulates e”.
- non-contextual fraction NCF(e): the largest λ with e = λe^NC + (1 − λ)e′, e^NC non-contextual; CF = 1 − NCF.
- Graham reduction: deleting a vertex that lies in exactly one maximal face; acyclic means reducible to the empty complex.
Connections
Builds on Abramsky and Brandenburger (LIT-016) for scenarios, models and the global-section criterion, and on the contextual fraction of Abramsky, Barbosa and Mansfield (LIT-265), which it shows to be functorial. The resource-theory reading follows Coecke, Fritz and Spekkens. The direct successor is the comonadic paper (LIT-847), which replaces the relation by a simplicial map plus adaptive measurement protocols and adds preprocessing; the chapter “Closing Bell” (LIT-845) revises the morphisms again (mixtures of deterministic procedures) and characterises which maps of models they induce.
Bearing on the record
- CLAIM-100 (directed transport between scenarios whose covers change is the theory’s open problem). This paper is prior art that CLAIM-100’s “Partial prior art” lacks. A morphism here is a directed, stochastic transport from one scenario to another whose cover may be entirely different: π assigns each target measurement a jointly measurable set of source measurements, and σ maps local outcomes stochastically. The paper proves what such transport preserves: non-contextuality is preserved and the contextual fraction cannot increase (Theorem 4.5), strong contextuality is reflected (4.3). So the “compatibility structure” half of A94’s question has a partial answer from 2018. What remains open, relative to CLAIM-100: transport of signalling data (excluded by Remark 3.2), and preservation of “decision-relevant information”, which no result here addresses. The simulation order is a resource order on contextuality, not an informativeness order.
- CLAIM-121 and CLAIM-070 (the manuscript’s Propositions 1 and 2). Both are special cases of results here, read by me rather than stated by the paper. A natural transformation σ is exactly a family of local kernels that commutes with restriction (CLAIM-121), and because it is defined on every subset of X, including X itself, it carries a global kernel σ_X whose restrictions are the local ones: CLAIM-070’s “one global stochastic kernel”. Theorem 4.1 composed with any simulation gives CLAIM-070’s conclusion, a global section of the target. Lemma 3.12 and the remark before it give CLAIM-070’s stated limit: local stochastic kernels need not glue to a global one, since D ∘ E is not a sheaf. The manuscript’s caution that “cover-changing … maps demand separate analysis” (§6) is answered for no-signalling models: π may change the cover arbitrarily so long as it is simplicial.
- CLAIM-105 and QUESTION-004 (transports that destroy or create contextuality). Destroying it is possible (any simulation to a non-contextual model). Creating it by free (classical) transport is impossible (Theorem 4.5).
- THEORY-156 (Blackwell: more informative iff garbling). A structural parallel the paper does not draw: a simulation must use one procedure for every target context, as a garbling uses one kernel for every state, and both orders are characterised by monotones. On a single-context scenario a simulation is a stochastic map of outcomes, a garbling of a one-state experiment. This is my observation; nothing here relates the simulation preorder to Blackwell’s.
- Produces THEORY-174, with its two successors.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- No-signalling models only (Remark 3.2).
- No preprocessing: randomised or adaptive choice of measurements is left open (§5); the comonadic paper supplies it.
- The monoidal structure is asserted to be symmetric monoidal without a written proof.
- Strong contextuality has no categorical characterisation here, only monotonicity (§4).
- No quantitative or computational content: whether a simulation d → e exists is not shown to be decidable or computable.
Open questions
- Whether other classes (strongly contextual, All-vs-Nothing, cohomologically obstructed, quantum-realisable) are characterised by the category (§5).
- Whether simulation with preprocessing (or with quantum resources) gives the “right” category; the author names the risk that combining a relation π with stochastic preprocessing is ill-defined (mixing x₁ with {x₂, x₃}).
- An analogue of complexity-theoretic completeness: is there a model that simulates every model of a fixed scenario’s contextuality class?