Contextuality and the fundamental theorems of quantum mechanics

Döring, Andreas; Frembs, Markus

doi:10.48550/arxiv.1910.09591

Cited by 2 publications

(2 citation statements)

References 48 publications

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“…As shown in Ref. [4] nonlocality is a special case of contextuality, and starting from the topos formalism for quantum theory, we have that contextuality is at the origin of all the main properties that make it what it is [24]. Therefore, contextuality is at the origin of what is known today as non-classicality and can be informally explained as the inability to a global agreement even if local agreement occurs [1].…”

Section: Introductionmentioning

confidence: 92%

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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Section: Introductionmentioning

confidence: 92%

Contextuality in the Fibration Approach and the Role of Holonomy

Montanhano¹

2021

Preprint

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“…Contextuality is proving to be the primary fuel for this revolution, as it is defined as the inability to be classical, at least in relation to a fixed notion of classicality. It is known that contextuality is the origin of quantum behavior [17], and it is the generalization of the famous notion of nonlocality [1]. It is necessary to any computational advantage on classical computers [18], and it is the "magic" necessary for some kinds of quantum computers [19].…”

Section: Introductionmentioning

confidence: 99%

Differential Geometry of Contextuality

Montanhano¹

2022

Preprint

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Contextuality has been related for a long time as a topological phenomenon. In this work, such a relationship is exposed in the more general framework of generalized contextuality. The main idea is to identify states, effects, and transformations as vectors living in a tangent space, and the non-contextual conditions as discrete closed paths implying null vertical phases. Two equivalent interpretations hold. The geometrical or realistic view, where flat space is imposed, implies that the contextual behavior becomes equivalent to the curvature (nontrivial holonomy) of the probabilistic functions, in analogy with the electromagnetic tensor; as a modification of the valuation function, it can be used to connect contextuality with interference, non-commutativity, and signed measures. The topological or anti-realistic view, where the valuation functions must be preserved, implies that the contextual behavior can be translated as topological failures (non-trivial monodromy); it can be used to connect contextuality with non-embeddability and a generalized Voroby'ev theorem. Both views can be related to contextual fraction, and the disturbance in ontic models can be presented as non-trivial transition maps.

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Contextuality and the fundamental theorems of quantum mechanics

Cited by 2 publications

References 48 publications

Contextuality in the Fibration Approach and the Role of Holonomy

Contextuality in the Fibration Approach and the Role of Holonomy

Differential Geometry of Contextuality

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