Effect-valued measurement models and contextuality

arXiv:2607.25900v1 Announce Type: new 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 \(\sigma:A\to[0,1]\) 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.

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 \(\sigma:A\to[0,1]\) 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.

arXiv:2607.25900 [quant-ph]

(or arXiv:2607.25900v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.25900 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

aiqbench analysis

Effect-valued measurement models and contextuality

Related coverage