The notion of contextuality, which emerges from a theorem established by Simon Kochen and Ernst Specker (1960-1967) and by John Bell (1964-1966), is certainly one of the most fundamental aspects of quantum weirdness. If it is a questioning on scholastic philosophy and a study of contrafactual logic that led Specker to his demonstration with Kochen, it was a criticism of von Neumann's "proof" that led John Bell to the result. A misinterpretation of this famous "proof" will lead them to diametrically opposite conclusions. Over the last decades, remarkable theoretical progresses have been made on the subject in the context of the study of quantum foundations and quantum information. Thus, the graphic generalizations of Cabello-Severini-Winter and Acin-Fritz-Leverrier-Sainz raise the question of the connection between non-locality and contextuality. It is also the case of the sheaf-theoretic approach of Samson Abramsky et al., which also invites us to compare contextuality with the logical structure of certain classical logical paradoxes. Another approach, initiated by Robert Spekkens, generalizes the concept to any type of experimental procedure. This new form of "universal" contextuality has been raised as a criterion of non-classicality, i.e. of weirdness. It notably led to identify the nature of curious quantum paradoxes involving post-selections and weak measurements. In the light of the fiftieth anniversary of the publication of the Kochen-Specker theorem, this report aims to introduce these results little known to the French scientific public, in the context of an investigation on the nature of the weirdness of quantum physics.
The notion of contextuality, which emerges from a theorem established by Simon Kochen and Ernst Specker (1960-1967) and by John Bell (1964-1966), is certainly one of the most fundamental aspects of quantum weirdness. If it is a questioning on scholastic philosophy and a study of contrafactual logic that led Specker to his demonstration with Kochen, it was a criticism of von Neumann's "proof" that led John Bell to the result. A misinterpretation of this famous "proof" will lead them to diametrically opposite conclusions. Over the last decades, remarkable theoretical progresses have been made on the subject in the context of the study of quantum foundations and quantum information. Thus, the graphic generalizations of Cabello-Severini-Winter and Acin-Fritz-Leverrier-Sainz raise the question of the connection between non-locality and contextuality. It is also the case of the sheaf-theoretic approach of Samson Abramsky et al., which also invites us to compare contextuality with the logical structure of certain classical logical paradoxes. Another approach, initiated by Robert Spekkens, generalizes the concept to any type of experimental procedure. This new form of "universal" contextuality has been raised as a criterion of non-classicality, i.e. of weirdness. It notably led to identify the nature of curious quantum paradoxes involving post-selections and weak measurements. In the light of the fiftieth anniversary of the publication of the Kochen-Specker theorem, this report aims to introduce these results little known to the French scientific public, in the context of an investigation on the nature of the weirdness of quantum physics.
Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful. We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches. We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations. This construction has a surprisingly broad range of uses. We apply it in the study of a number of issues, including: - establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras; - formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen-Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics; - investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations; - developing some work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.
A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here, compatible measurements are those that can be performed simultaneously or in any order without disturbance. This conflict is generically called quantum contextuality. In this article, we present an introduction to this subject and its current status. We review several proofs of the Kochen-Specker theorem and different notions of contextuality. We explain how to experimentally test some of these notions and discuss connections between contextuality and nonlocality or graph theory. Finally, we review some applications of contextuality in quantum information processing.
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.
Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr [Albert Einstein: Philosopher-Scientist, Library of Living Philosophers Vol. VII, edited by P. A. Schilpp (Open Court, 1998), pp. 199–241] and later realized more technically by Kochen and Specker [J. Math. Mech. 17, 59 (1967)]. 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 Isham and Butterfield [Int. J. Theor. Phys. 37, 2669 (1998)]. Here, we broaden this perspective considerably (partly drawing on existing, but scattered results) and show that apart from the Kochen–Specker theorem, Wigner’s theorem, Gleason’s theorem, and Bell’s theorem also 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.
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.
Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that an important class of contextuality arguments has a topological origin. More specifically, we show that "All-vs-Nothing" proofs of contextuality are witnessed by cohomological obstructions.