The Kochen-Specker theorem is a basic and fundamental 50 year old non-existence result affecting the foundations of quantum mechanix, strongly implying the lack of any meaningful notion of "quantum realism", and typically leading to discussions of "contextuality" in quantum physics. Original proofs of the Kochen-Specker theorem proceeded via brute force counter-examples; often quite complicated and subtle (albeit mathematically "elementary") counter-examples. Only more recently have somewhat more "geometrical" proofs been developed. We present herein yet another simplified geometrical proof of the Kochen-Specker theorem, one that is valid for any number of dimensions, that minimizes the technical machinery involved, and makes the seriousness of the issues raised manifest.
The Kochen-Specker theorem is a basic and fundamental 50 year old non-existence result affecting the foundations of quantum mechanix, strongly implying the lack of any meaningful notion of "quantum realism", and typically leading to discussions of "contextuality" in quantum physics. Original proofs of the Kochen-Specker theorem proceeded via brute force counter-examples; often quite complicated and subtle (albeit mathematically "elementary") counter-examples. Only more recently have somewhat more "geometrical" proofs been developed. We present herein yet another simplified geometrical proof of the Kochen-Specker theorem, one that is valid for any number of dimensions, that minimizes the technical machinery involved, and makes the seriousness of the issues raised manifest.
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 central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics cannot be reconciled with classical models that are noncontextual for ideal measurements. The first explicit derivation by Kochen and Specker was rather complex, but considerable simplifications have been achieved thereafter. We propose a systematic approach to find minimal Hardy-type and Greenberger-Horne-Zeilinger-type (GHZ-type) proofs of the Kochen-Specker theorem, these are characterized by the fact that the predictions of classical models are opposite to the predictions of quantum mechanics. Based on our results, we show that the Kochen-Specker set with 18 vectors from Cabello et al. [Phys. Lett. A 212, 183 (1996)PYLAAG0375-960110.1016/0375-9601(96)00134-X] is the minimal set for any dimension, verifying a longstanding conjecture by Peres. Our results allow to identify minimal contextuality scenarios and to study their usefulness for information processing.
The purpose of this note is to complete the interesting review on quantum contextuality [1] that appeared recently. In particular we will introduce and discuss the ideas of extracontextuality and extravalence, that allow one to relate Kochen-Specker's and Gleason's theorems, and also to shift the emphasis from the first to the second one. We will also argue that whereas Kochen-Specker's is essentially a negative result (a no-go theorem), Gleason's is a positive one since it provides a mathematical justification of Born's rule. The link between these issues is provided by a specific quantum feature that we call extravalence.
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.
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.
In this work we begin with a critical review of the concept of observable in a generical physical theory; relying on C*-algebras formalism and on Gelfand-Naimark theorem we
justify the link between observables and operators in an Hilbert space. We discuss particularly observables of quantum mechanics, introducing some mathematical results that characterizes them. We introduce the problem of hidden variables and the hypothesis of non-contextuality, than we give a proof of the Kochen-Specker theorem in dimension greater or equal to 3 using partial algebras formalism; we also show a constructive example of an hidden variable theory in dimension 2. Finally we discuss Meyer’s and Kent’s objections relying on finite precision of physical measurements and we present colorations of S² ∩ Q³ and of dense subsets of the projectors of C^n and l²(C) that respect Kochen-Specker rules.