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arXiv:1709.04812

On geometrical aspects of the graph approach to contextuality

Bárbara Amaral, Marcelo Terra Cunha

Published Sep 14, 2017• 5 citations
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Abstract

The connection between contextuality and graph theory has led to many developments in the field. In particular, the sets of probability distributions in many contextuality scenarios can be described using well known convex sets from graph theory, leading to a beautiful geometric characterization of such sets. This geometry can also be explored in the definition of contextuality quantifiers based on geometric distances, which is important for the resource theory of contextuality, developed after the recognition of contextuality as a potential resource for quantum computation. In this paper we review the geometric aspects of contextuality and use it to define several quantifiers, which have the advantage of being applicable to the exclusivity approach to contextuality, where previously defined quantifiers do not fit.

DOI
10.48550/arxiv.1709.04812

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The connection between contextuality and graph theory has led to many developments in the field. In particular, the sets of probability distributions in many contextuality scenarios can be described using well known convex sets from graph theory, leading to a beautiful geometric characterization of such sets. This geometry can also be explored in the definition of contextuality quantifiers based on geometric distances, which is important for the resource theory of contextuality, developed after the recognition of contextuality as a potential resource for quantum computation. In this paper we review the geometric aspects of contextuality and use it to define several quantifiers, which have the advantage of being applicable to the exclusivity approach to contextuality, where previously defined quantifiers do not fit.
The connection between contextuality and graph theory has paved the way for numerous advancements in the field. One notable development is the realization that sets of probability distributions in many contextuality scenarios can be effectively described using well-established convex sets from graph theory. This geometric approach allows for a beautiful characterization of these sets. The application of geometry is not limited to the description of contextuality sets alone; it also plays a crucial role in defining contextuality quantifiers based on geometric distances. These quantifiers are particularly significant in the context of the resource theory of contextuality, which emerged following the recognition of contextuality as a valuable resource for quantum computation. In this paper, we provide a comprehensive review of the geometric aspects of contextuality. Additionally, we use this geometry to define several quantifiers, offering the advantage of applicability to other approaches to contextuality where previously defined quantifiers may not be suitable. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.
In addition to the important role of contextuality in foundations of quantum theory, this intrinsically quantum property has been identified as a potential resource for quantum advantage in different tasks. It is thus of fundamental importance to study contextuality from the point of view of resource theories, which provide a powerful framework for the formal treatment of a property as an operational resource. In this contribution, we review recent developments towards a resource theory of contextuality and connections with operational applications of this property. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.
Quantum theory departs from classical physics in its treatment of correlations, most prominently through the phenomena of contextuality and nonlocality. Once regarded primarily as foundational curiosities, these effects are now understood as key operational resources for quantum computation, communication, and simulation. Although traditionally investigated in distinct settings, recent theoretical and experimental advances have revealed deep conceptual, mathematical, and operational connections between them. This review presents a unified perspective on these developments based on sheaf-theoretic and graph-theoretic frameworks, which provide theory-independent characterizations of statistical correlations. These approaches clarify the structural relationship between contextuality and nonlocality, facilitate the formulation of experimentally testable inequalities, and guide implementations in realistic physical platforms, with particular emphasis on photonic systems. By bridging abstract theoretical structures and concrete experimental realizations, this review sheds light on the nonclassical foundations of quantum correlations and their emerging role in quantum technologies.
In this paper, we explore the graph approach to contextuality to restate the extended definition of noncontextuality as given by Kujala et al. [Phys. Rev. Lett. 115, 150401 (2015)] using graph-theoretical terms. This extended definition avoids the assumption of the pre-sheaf or non-disturbance condition, which states that if two contexts overlap, then the marginal distribution obtained for the intersection must be the same, a restriction that will never be perfectly satisfied in real experiments. With this, we are able to derive necessary conditions for extended noncontextuality for any set of random variables based on the geometrical aspects of the graph approach, which can be tested directly with experimental data in any contextuality experiment and which reduce to traditional necessary conditions for noncontextuality if the non-disturbance condition is satisfied.
In this thesis, we explore the intersection of two fundamental subfields of quantum information theory: quantum coherence and contextuality. Despite their apparent differences, both areas address key issues relevant to the foundations and applications of quantum theory. By developing a novel graph-theoretic approach, extending a framework recently introduced by Galvão and Brod (Phys. Rev. A 101, 062110, 2020), we establish a formal connection between inequality-based witnesses of quantum coherence and noncontextuality inequalities. Our key contributions include: the development of a graph-theoretic framework for generating coherence and contextuality witnesses; a formal mapping between the inequalities that follows from the work by Galvão and Brod to existing noncontextuality inequalities; the conceptualization of a relational form of quantum coherence; a proof of contextual advantage for the task of quantum interrogation; and the discovery of an infinite family of coherence witnesses that also require quantum states in Hilbert spaces of specific dimensions.
In an ideal contextuality scenario it is assumed that, whenever two contexts overlap, the marginal distribution obtained for the intersection of these contexts must be the same. Although this nondisturbance idealization has been extremely useful in investigating the limitations of physical theories, in this paper, we will work out a more robust and possibly more experimentally friendly framework for contextuality. Simply put, we will restate the extended definition of noncontextuality provided by the noncontextuality by default and show that in this extended perspective it is possible to avoid weakening the nondisturbance hypothesis. By doing so, we also show how standard tools for characterizing and quantifying contextuality can also be applied to this extended scenario. In particular, we came up with a linear program to check whether or not a behavior is contextual in the extended sense. The connection with the usual definition of contextuality is also established because whenever nondisturbance is fully satisfied the extended conditions reduce to the original ones.