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arXiv:2408.16764

An algebraic characterisation of Kochen-Specker contextuality

Markus Frembs

Published Aug 29, 2024• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.2408.16764

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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 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.
We address the problem of deriving the set of quantum correlations for every Bell and Kochen-Specker (KS) contextuality scenario from simple assumptions. We show that the correlations that are possible according to quantum theory are equal to those possible under the assumptions that there is a nonempty set of correlations for every KS scenario and a statistically independent realization of any two KS experiments. The proof uses tools of the graph-theoretic approach to correlations and deals with Bell nonlocality and KS contextuality in a unified way.
The Kochen–Specker theorem establishes that noncontextual, globally consistent assignments of definite values to quantum observables are mathematically impossible. While this result places a fundamental constraint on pre-measurement value assignments, it does not by itself specify why observable values should exhibit contextuality or how the pre-measurement domain should be conceptually understood. This paper examines the Kochen–Specker theorem from the perspective of the Intrinsic Observability Framework (IOF), with particular emphasis on the relationship between contextuality and accessibility. In the IOF, observable values are not regarded as pre-existing entities awaiting discovery. Instead, the pre-measurement domain is characterized by potential observability, and measurement grants accessibility, thereby establishing observability and making observable values operationally meaningful. From this perspective, different measurement contexts generally participate in distinct accessibility conditions. Even when the same observable appears in distinct compatible sets, the conditions under which its observable value is operationally established are not necessarily identical. Contextuality can therefore be understood not merely as a restriction on global value assignment, but as a natural consequence of accessibility-dependent observability. The Kochen–Specker impossibility of constructing a single noncontextual global value map is thus interpreted as reflecting the attempt to combine observable values established under distinct accessibility conditions into a context-independent description. The IOF does not modify the mathematical content of the Kochen–Specker theorem or the standard quantum formalism. Rather, it provides an accessibility-based conceptual account of the contextual structure revealed by the theorem while explicitly characterizing the pre-measurement domain. Finally, the relationship between the IOF and representative interpretations of quantum mechanics is compared to clarify the distinct conceptual role of accessibility-dependent observability within quantum foundations.
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.
Quantum contextuality is a concept used to describe the property of hidden-variable theory that measurement outcomes predetermined by the hidden variables depend on the measurement context. The term measurement context can have different meanings, giving rise to different flavours of quantum contextuality. The first discovered flavour is Kochen-Specker (KS) contextuality where measurement outcomes will depend on what compatible measurements are jointly performed with the selected measurement. Another flavour, here to be compared with KS contextuality, is that referred to in Bohmian mechanics where outcomes of some specific measurements are not completely specified by the model state, but depend also on specifics of the measurement device used. It has been claimed that this type of Bohmian contextuality is necessary to enable KS contextuality in a hidden variable model. In this paper we show that this is not the case. The recently proposed Contextual Ontological Model (COM) [Hindlycke and Larsson, Phys. Rev. Lett. 2022] produces KS contextual predictions but does not have the Bohmian contextuality; the outcome of every measurement allowed by COM can be predicted from the model state itself. This distinguishes Bohmian contextuality from KS contextuality, and enables individual study of the two concepts.
Contextuality is a key feature of quantum information that challenges classical intuitions, providing the basis for constructing explicit proofs of quantum advantage. While a number of evidences of quantum advantage are based on the contextuality argument, the definition of contextuality is different in each research, causing incoherence in the establishment of instant connection between their results. In this report, we review the mathematical structure of sheaf-theoretic contextuality and extend this framework to explain Kochen-Specker type contextuality. We first cover the definitions in contextuality with detailed examples. Then, we state the all-versus-nothing (AvN) argument and define a state-independent AvN class. It is shown that Kochen-Specker type contextuality, or contextuality in a partial closure, can be translated into this framework by the partial closure of observables under the multiplication of commuting measurements. Finally, we compare each case of contextuality in an operator-side view, where the strict hierarchy of contextuality class in a state-side view seems to merge into the state-independent AvN class together with the partial closure formalism. Overall, this report provides a unified interpretation of contextuality by integrating Kochen-Specker type notions into the state-independent AvN argument. The results present novel insights into contextuality, which pave the way for a coherent approach to constructing proofs of quantum advantage.