Effect-valued measurement models and contextuality

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.

Source: arXiv:2607.25900v1 - http://arxiv.org/abs/2607.25900v1 PDF: https://arxiv.org/pdf/2607.25900v1 Original Link: http://arxiv.org/abs/2607.25900v1