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 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.
Quantum coherence, nonlocality, and contextuality are key resources for quantum advantage in metrology, communication, and computation. We introduce a graph-based approach to derive classicality inequalities that bound local, noncontextual, and coherence-free models, offering a unified description of these seemingly disparate quantum resources. Our approach generalizes recently proposed basis-independent coherence witnesses, and it recovers all noncontextuality inequalities of the exclusivity graph approach. Moreover, violations of certain classicality inequalities witness preparation contextuality. We describe an algorithm to find all such classicality inequalities, and we use it to analyze some of the simplest scenarios.
We analyse nonclassical resources in interference phenomena using generalized noncontextuality inequalities and basis-independent coherence witnesses. We use recently proposed inequalities that witness both resources within the same framework. We also propose, in view of previous contextual advantage results, a systematic way of applying these tools to characterize advantage provided by coherence and contextuality in quantum information protocols. We instantiate this methodology for the task of quantum interrogation, famously introduced by the paradigmatic bomb-testing interferometric experiment, showing contextual quantum advantage for such a task.
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.
The main subjects of this thesis are non-locality and contextuality, two fundamental features of quantum mechanics that constitute valuable resources for quantum computation. Our analysis is based on Abramsky & Brandenburger's sheaf theoretic framework, which captures both these phenomena in a unified treatment and in a very general setting. This high-level description transcends quantum physics and allows to precisely characterise the notion of contextuality as the apparent paradox realised by data being locally consistent but globally inconsistent. More specifically, we aim to develop a deeper understanding of so-called logical forms of contextuality, i.e. situations where the phenomenon can be witnessed using purely logical arguments, disregarding probabilities. The sheaf theoretic description of logical contextuality has recently inspired the development of a topological treatment of the phenomenon based on sheaf cohomology. In this thesis, we embark on a detailed analysis of the cohomology of contextuality, exposing key shortcomings in the current methods, and introducing an (almost) complete cohomological characterisation of logical forms of contextuality. More specifically, we show that, in its current formulation, sheaf cohomology does not constitute a complete invariant for contextuality, not even in its strongest forms, and that higher cohomology groups cannot be used to study the phenomenon. Then, we solve these issues by introducing a novel construction, which derives refined versions of the presheaves describing empirical models to expose their deeper extendability properties, resulting in a sheaf cohomological invariant which is applicable to the vast majority of empirical models, and conjectured to work universally. We propose a general theory of contextual semantics using the language of valuation algebras. In particular, we give a general definition of contextual behaviour as a fundamental gap between local agreement and global disagreement of information sources. Not only does this formalism aptly capture and generalise the known instances of contextuality beyond quantum theory, but it also provides inspiration for further applications of the phenomenon, and paves the way for the transfer of results and techniques between different fields. We give a prime example of this potential by developing faster algorithms to detect contextuality based on mainstream methods of generic inference. Finally, we turn our attention back to instances of contextuality in quantum physics, and study strong contextuality in multi-qubit states. We give a complete combinatorial characterisation of All-vs-Nothing proofs of strong contextuality in stabiliser quantum mechanics. This allows to produce the complete list of all stabiliser states exhibiting this kind of contextuality, which consitututes an important resource in certain models of quantum computation. Then, we extend our search for strongly contextual behaviour beyond stabiliser states, and identify the minimum quantum resources needed to realise strong non-locality. Additional results include a partial classification of strongly non-local models comprised of three-qubit states and local projective measurements, and the introduction of a new infinite family of strongly non-local three-qubit states.
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'.
Contextuality is a key distinguishing feature between classical and quantum physics. It expresses a fundamental obstruction to describing quantum theory using classical concepts. In turn, understood as a resource for quantum computation, it is expected to hold the key to quantum advantage. Yet, despite its long recognised importance in quantum foundations and, more recently, in quantum computation, the structural essence of contextuality has remained somewhat elusive - different frameworks address different aspects of the phenomenon, yet their precise relationship often remains unclear. This issue already looms large at the level of the Bell-Kochen-Specker theorem: while traditional proofs proceed by showing the nonexistence of valuations, the notion of state-independent contextuality in the marginal approach allows to prove contextuality from seemingly weaker assumptions. In the light of this, and at the absence of a unified mathematical framework for Kochen-Specker contextuality, the original algebraic approach has been widely abandoned, in favour of the study of contextual correlations. Here, we reinstate the algebraic perspective on contextuality. Concretely, by building on the novel concept of context connections, we reformulate the algebraic relations between observables originally postulated by Kochen and Specker, and we explicitly demonstrate their consistency with the notion of state-independent contextuality. In the present paper, we focus on the new conceptual ideas and discuss them in the concrete setting of spin-1 observables, specifically those in the example of [S. Yu and C.H. Oh, Phys. Rev. Lett., 108, 030402 (2012)]; in a companion paper, we generalise these ideas, obtain a complete characterisation of Kochen-Specker contextuality and provide a detailed comparison with the related notions of contextuality in the marginal and graph-theoretic approach.