Different analytic notions of contextuality fall into two major groups: probabilistic and strong notions of contextuality. Kochen and Specker's Theorem 0 [1] presents a demarcation criterion for differentiating between those groups. Whereas probabilistic contextuality still allows classical models, albeit with nonclassical probabilities, the logico-algebraic “strong” form of contextuality characterizes collections of quantum observables that have no faithfully embedding into (extended) Boolean algebras. Both forms indicate a classical in- or under-determination that can be termed “value indefinite” and formalized by partial functions of theoretical computer sciences.
Different analytic notions of contextuality fall into two major groups: probabilistic and strong notions of contextuality. Kochen and Specker's Theorem 0 [1] presents a demarcation criterion for differentiating between those groups. Whereas probabilistic contextuality still allows classical models, albeit with nonclassical probabilities, the logico-algebraic “strong” form of contextuality characterizes collections of quantum observables that have no faithfully embedding into (extended) Boolean algebras. Both forms indicate a classical in- or under-determination that can be termed “value indefinite” and formalized by partial functions of theoretical computer sciences.
Contextuality can either be synthetically defined in terms of outcome conditionality on the measurement conditions, or in terms of non-classical probability distributions. Another logico-algebraic strong form of contextuality characterizes collections of quantum observables that have no faithfully embedding into (extended) Boolean algebras. Any of these forms indicate a classical in- or underdetermination that can be termed value indefinite, and formalized by partial functions of theoretical computer sciences. he term contextual by indeterminate or value indefinite in the spirit of partial functions of theoretical computer sciences.
The notion of in terms of nonclassical (paradoxical) properties of probability distributions is refined by concentrating on the nonclassical logico-algebraic structures of the observables supporting such probabilities. These structures often, but not always, induce and are reflected by nonclassical properties of the probability distributions. Two types of contextuality are introduced, originating either in nonclassical probabilities on classical logics, or in genuine nonclassical logics of observables which cannot be faithfully (homomorphically) embedded into (extended) Boolean algebras. It is suggested to substitute the term contextual by indeterminate or value indefinite in the spirit of partial functions of theoretical computer sciences.
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.
Abstract Quantum contextuality represents a fundamental form of nonclassicality in quantum mechanics. To provide a more complete characterization of nonclassical properties in quantum systems, we adopt a logical perspective and propose a mathematical framework based on exclusive partial Boolean algebras. This framework enables a unified description of contextuality and nonclassicality across finite general, quantum, and classical systems. We establish a unified and minimal classical counterpart for any finite general system. Within this framework, we formalize major categories of quantum contextuality, demonstrating that: 12 projectors suffice to generate Kochen–Specker scenarios; 10 projectors suffice to witness state-independent contextuality; and 3 observables suffice to witness quantum contextuality. Finally, we prove that contextuality is a sufficient but not necessary condition for nonclassicality.
In the literature, there are two differing definitions of contextuality: Kochen and Specker's, and Spekkens' (or ``generalised''). However, researchers using one of these definitions rarely consider the other, meaning comparative analysis of these two notions is rare. In this paper, we advance the idea that Kochen-Specker contextuality provides a generalisation of the idea of system being fundamentally \emph{nonclassical}, while Spekkens' noncontextuality provides a generalisation of the idea of a system being \emph{classical}. This allows us to reconcile the two approaches, as different stages in a hierarchy of classicality/nonclassicality.
Since Bell, Kochen and Specker formulated the first notions of contextuality, a number of frameworks have been created to help make philosophical and mathematical sense of the concept. Two of the most common are the Equivalence-based approach due to Spekkens, and the Sheaf-theoretic framework of Abramsky and Brandenburger. Each of these can support a hierarchy of strengths of contextuality, which can include possibilistic concepts. This thesis will explore the differences between the two approaches, and explore what can be demonstrated by using weaker notions of noncontextuality. Notions of noncontextuality are important for proving no-go theorems which highlight that many properties that are true of classical mechanics fail to be compatible with quantum predictions. Often, the notions of noncontextuality used are very strong assumptions, and even weaker flavours of noncontextuality can be used to draw the same, or similar conclusions. An important class of such notions are, possibilistic assumptions that only care about whether or not events are possible or impossible, rather than the exact Born rule probabilities that are applied to events. Even these weaker notions are often impossible to resolve with the logical structure and predictions of quantum theory. Contextuality has been demonstrated to be a resource for many quantum computational operations, and the contextual fraction has been shown to be an important resource. We introduce a metric that can differentiate between different quantum mechanical scenarios with contextual fraction 1, and demonstrate analogues of CHSH and Tsirelson bounds for this quantity. We also investigate the computational complexity of calculating the values of some metrics for nonlocality and contextuality, completing the classification of possibilistic nonlocality scenarios, as well as other metrics of nonlocality analogous to our metric of maximal contextuality.