Contextuality and bundle diagrams

Beer, Kerstin; Osborne, Tobias J.

doi:10.1103/physreva.98.052124

Cited by 10 publications

(7 citation statements)

References 44 publications

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“…For MBQC, a classification of contextuality in terms of group cohomology has been given in [52,47]. Similar connections between contextuality and cohomology have also been obtained in [53] and within the sheaf theoretic formalism in [3,7,12]. The latter framework provides a precise threshold for contextual computation by means of the contextual fraction [1,48].…”

Section: Discussionmentioning

confidence: 76%

“…of characters (or pure states) of V , traditionally called the Gelfand spectrum of V . 7 In physical terms, Σ(V ) is the (pure) state space of the physical system described by the physical quantities in V . As expected, the points of the state space Σ(V ) correspond exactly with valuation functions on V sa .…”

Section: The Spectral Presheaf and The Reformulation Of The Kochen-sp...mentioning

confidence: 99%

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Contextuality and the fundamental theorems of quantum mechanics

Döring¹,

Frembs²

2019

Preprint

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Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr [11] and later realised more technically by Kochen and Specker [43]. Isham and Butterfield put contextuality at the heart of their topos-based formalism and gave a reformulation of the Kochen-Specker theorem in the language of presheaves in [37].Here, we broaden this perspective considerably (partly drawing on existing, but scattered results) and show that apart from the Kochen-Specker theorem, also Wigner's theorem, Gleason's theorem and Bell's theorem relate fundamentally to contextuality. We provide reformulations of the theorems using the language of presheaves over contexts and give general versions valid for von Neumann algebras. This shows that a very substantial part of the structure of quantum theory is encoded by contextuality.

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Section: Discussionmentioning

confidence: 76%

Section: The Spectral Presheaf and The Reformulation Of The Kochen-sp...mentioning

confidence: 99%

Contextuality and the fundamental theorems of quantum mechanics

Döring¹,

Frembs²

2019

Preprint

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“…The possibilistic structure of a behaviour can be visualized in its bundle diagram [35]. These consists of a base space formed by the compatibility hypergraph of the scenario (see Fig.…”

Section: Logical Contextualitymentioning

confidence: 99%

On Possibilistic Conditions to Contextuality and Nonlocality

Santos,

Amaral

2020

Preprint

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Contextuality and non-locality are non-classical properties exhibited by quantum statistics whose implications have profound impact on the foundations and applications of quantum theory. In this paper we provide some insights into what is called logical contextuality, which is, in a sense, a possibilistic version of contextuality. In particular, we demonstrate that the occurrence of a certain set of possibilistic conditions, which we call a Hardy paradox, is necessary and sufficient for logical contextuality.

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“…Thus, contextuality appears as a property of the bundle, as a total space. Its use already appeared as a graphic resource in the sheaf approach [10], but its due formalism occurred later in Ref. [19].…”

Section: Introductionmentioning

confidence: 99%

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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Contextuality and bundle diagrams

Cited by 10 publications

References 44 publications

Contextuality and the fundamental theorems of quantum mechanics

Contextuality and the fundamental theorems of quantum mechanics

On Possibilistic Conditions to Contextuality and Nonlocality

Contextuality in the Fibration Approach and the Role of Holonomy

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