Fully revealing the mathmatical structure of quantum contextuality is a significant task, while some known contextuality theories are only applicable for rank-1 projectors. That is because they adopt the observable-based definitions. This paper overcomes the challenges faced by some known contextuality theories by establishing an event-based contextuality theory with exclusive partial Boolean algebra, which is used to describe the contextual systems with local consistency and exclusivity principle. Our theory provides a precise mathematical framework for quantum contextuality, which can handle the scenarios composed of general projectors, and introduces a more complete contextuality hierarchy. We conclude that the Kochen-Specker contextuality is equivalent to the state-independent strong contextuality for finite dimensional quantum systems. Therefore, when considering both the strength and proportion of contextual quantum states, Kochen-Specker contextuality is the strongest.
Fully revealing the mathmatical structure of quantum contextuality is a significant task, while some known contextuality theories are only applicable for rank-1 projectors. That is because they adopt the observable-based definitions. This paper overcomes the challenges faced by some known contextuality theories by establishing an event-based contextuality theory with exclusive partial Boolean algebra, which is used to describe the contextual systems with local consistency and exclusivity principle. Our theory provides a precise mathematical framework for quantum contextuality, which can handle the scenarios composed of general projectors, and introduces a more complete contextuality hierarchy. We conclude that the Kochen-Specker contextuality is equivalent to the state-independent strong contextuality for finite dimensional quantum systems. Therefore, when considering both the strength and proportion of contextual quantum states, Kochen-Specker contextuality is the strongest.
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.
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.
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.
A set of quantum measurements exhibits quantum contextuality when any consistent value assignment to the measurement outcomes leads to a contradiction with quantum theory. In the original Kochen-Specker-type of argument the measurement projectors are assumed to be rays, that is, of unit rank. Only recently a contextuality scenario has been identified where state-independent contextuality requires measurements with projectors of rank two. Using the disjunctive graph product, we provide a systematic method to construct contextuality scenarios which require non-unit rank. We construct explicit examples requiring ranks greater than rank one up to rank five.
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, when 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 mathematics of contextuality has remained somewhat elusive - different frameworks address different aspects of the phenomenon, yet their precise relationship often is unclear. In fact, there is a glaring discrepancy already between the original notion of contextuality introduced by Kochen and Specker on the one side [J. Math. Mech., 17, 59, (1967)], and the modern approach of studying contextual correlations on the other [Rev. Mod. Phys., 94, 045007 (2022)]. In a companion paper [arXiv:2408.16764], we introduce the conceptually new tool called ``context connections'', which allows to cast and analyse Kochen-Specker (KS) contextuality in new form. Here, we generalise this notion, and based on it prove a complete characterisation of KS contextuality for finite-dimensional systems. To this end, we develop the framework of ``observable algebras". We show in detail how this framework subsumes the marginal and graph-theoretic approaches to contextuality, and thus that it offers a unified perspective on KS contextuality. In particular, we establish the precise relationships between the various notions of ``contextuality" used in the respective settings, and in doing so, generalise a number of results on the characterisation of the respective notions in the literature.
Strong O-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory
Gleason's theorem identifies the Born rule via non-contextuality over an infinite continuous lattice of projectors, while its corollary the Kochen-Specker (KS) theorem rules out the specific class of deterministic ({0,1}-valued) noncontextual models using finite sets of projectors. Gleason's theorem also indicates the existence of KS-type finite vector constructions to rule out other discrete nondeterministic alternatives to quantum theory beyond the {0,1} case. Here, we construct finite measurement configurations to rule out noncontextual empirical models with outcome probabilities drawn from an arbitrary finite subset O⊂[0,1]. We do this by two means: (i) constructing a family of experimentally feasible state-dependent Hardy-type tests, and (ii) proving a generalized KS theorem for a broad class of O that includes prior results as special cases. In the sheaf-theoretic framework of Abramsky and Brandenburger, a hierarchy of probabilistic-possibilistic-strong contextuality has been established quantifying contextuality as a resource. We extend this framework by introducing strong O-valued contextuality, showing that quantum theory evades global sections of all finite-valued presheaves. We also discuss the implications of the result on finite many-valued logics as viable ontological models for quantum theory and for contextuality-based (semi)-device-independent protocols.