Effect-valued measurement models and contextuality

• URL: http://arxiv.org/abs/2607.25900v1
• Date: Tue, 28 Jul 2026 15:58:56 GMT
• Title: Effect-valued measurement models and contextuality
• Abstract summary: We generalize the Abramsky--Brandenburger sheaf-theoretic treatment of contextuality by replacing probability distributions with distributions valued in a convex effect algebra (A)<n>This yields a notion of (A)-valued measurement model encompassing probabilistic, deterministic, and quantum measurement structures.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We generalize the Abramsky--Brandenburger sheaf-theoretic treatment of contextuality by replacing probability distributions with distributions valued in a convex effect algebra \(A\). This yields a notion of \(A\)-valued measurement model encompassing probabilistic, deterministic, and quantum measurement structures within a single framework. States \(σ:A\to[0,1]\) induce ordinary empirical models, allowing observable behaviour to be viewed as arising from effect-valued measurement data. This leads to a distinction between internal contextuality of \(A\)-models and observable contextuality after state evaluation. We analyze the relationship between these notions and identify coherence conditions under which observable classical explanations assemble into internal ones. Using the ordered-vector-space representation of effect modules, we show that non-contextuality is characterized by feasibility of an associated cone program, generalizing the linear-programming formulation of contextuality in the probabilistic case. We also introduce an effect-valued contextual fraction and study its relation to observable contextuality witnesses. We show how cone duality leads to a notion of Bell witnesses for contextuality as a direct generalization of Bell inequalities. Finally, we analyze sharp realizability and uniform dilation for measurement models, clarifying the relationship between general POVMs and projective measurements, and show how resource-indexed effect structures induce graded monads generalizing the quantum monad.

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