Categories of Empirical Models

Karvonen, Martti

doi:10.4204/eptcs.287.14

Cited by 11 publications

(12 citation statements)

References 21 publications

(30 reference statements)

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“…One can then consider ways of extending such simulations. In [10] more expressive power was obtained by allowing π(x) to be a set of measurements instead of a singleton, and by allowing outcomes to be interpreted stochastically.…”

Section: The Categorical Viewpointmentioning

confidence: 99%

“…Using rules ( 14)-( 18), all uses of probabilistic mixing can be taken to the top level. By the kind of associativity rule (10), t can then be rewritten to a term of the form…”

Section: Equational Theorymentioning

confidence: 99%

“…• On the simulation side, we enhance the treatment given in [10] by introducing a measurement protocols construction on empirical models (Section IV-B). Measurement protocols were first introduced in a different setting in [12].…”

Section: Introductionmentioning

confidence: 99%

“…In [10], the other present author introduced a notion of simulation between (possibly contextual) behaviours, as morphisms between empirical models, in the setting of the "sheaf-theoretic" approach to contextuality introduced in [11]. This established a basis for a simulation-based approach to comparing resources.…”

Section: Introductionmentioning

confidence: 99%

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A comonadic view of simulation and quantum resources

Abramsky

Barbosa

Karvonen

et al. 2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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We study simulation and quantum resources in the setting of the sheaf-theoretic approach to contextuality and nonlocality. Resources are viewed behaviourally, as empirical models. In earlier work, a notion of morphism for these empirical models was proposed and studied. We generalize and simplify the earlier approach, by starting with a very simple notion of morphism, and then extending it to a more useful one by passing to a co-Kleisli category with respect to a comonad of measurement protocols. We show that these morphisms capture notions of simulation between empirical models obtained via "free" operations in a resource theory of contextuality, including the type of classical control used in measurement-based quantum computation schemes.

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Section: The Categorical Viewpointmentioning

confidence: 99%

“…Using rules ( 14)-( 18), all uses of probabilistic mixing can be taken to the top level. By the kind of associativity rule (10), t can then be rewritten to a term of the form…”

Section: Equational Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A comonadic view of simulation and quantum resources

Abramsky

Barbosa

Karvonen

et al. 2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…As obtained in Ref. [30], being acyclic induces morphisms -there linked with simplification of the object -from the terminal object of the category of empirical models.…”

Section: Necessary Condition To Contextual Behaviormentioning

confidence: 74%

Contextuality in the Fibration Approach and the Role of Holonomy

Montanhano¹

2021

Preprint

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Contextuality can be understood as the impossibility to construct a globally consistent description of a model even if there is local agreement. In particular, quantum models present this property. We can describe contextuality with the fibration approach, where the scenario is represented as a simplicial complex, the fibers being the sets of outcomes, and contextuality as the non-existence of a global section in the measure fibration, allowing direct representation and formalization of the already used bundle diagrams. Using the generalization to continuous outcome fibers, we built the concept of measure fibration, showing the Fine-Abramsky-Brandenburger theorem for the fibration formalism in the case of non-finite fibers. By the Voroby'ev theorem, we argue that the dependence of contextual behavior of a model to the topology of the scenario is an open problem. We introduce a hierarchy of contextual behavior to explore it, following the construction of the simplicial complex. GHZ models show that quantum theory has all levels of the hierarchy, and we exemplify the dependence on higher homotopical groups by the tetraedron scenario, where non-trivial topology implies an increase of contextual behavior for this case. For the first level of the hierarchy, we construct the concept of connection through Markov operators for the measure bundle using the measure on fibers of contexts with two measurements and taking the case of equal fibers we can identify the outcome space as the basis of a vector space, that transform according to a group extracted from the connection. With this, it is possible to show that contextuality at the level of contexts with two measurements has a relationship with the non-triviality of the holonomy group in the frame bundle. We give examples and treat disturbing models through transition functions, generalizing the holonomy.

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