The Kochen–Specker theorem shows that it is impossible to assign sharp values to all dynamical variables in quantum mechanics in such a way that the algebraic relations among the values of dynamical variables whose self-adjoint operators commute are the same as those among the operators themselves. We point out that, for quantum theories obtained by quantizing some classical theory, this condition – Kochen–Specker non-contextuality – is implausible from the start because quantization usually changes algebraic relations. We explain why this is so, using the formalism of deformation quantization and its conception of star products , and we illustrate the relevance of this point using various examples of dynamical variables quantized via Weyl quantization and coherent state quantization. Our observations suggest that the relevance of the Kochen–Specker theorem to the question of whether one can assign sharp values to all dynamical variables is rather limited.
The Kochen–Specker theorem shows that it is impossible to assign sharp values to all dynamical variables in quantum mechanics in such a way that the algebraic relations among the values of dynamical variables whose self-adjoint operators commute are the same as those among the operators themselves. We point out that, for quantum theories obtained by quantizing some classical theory, this condition – Kochen–Specker non-contextuality – is implausible from the start because quantization usually changes algebraic relations. We explain why this is so, using the formalism of deformation quantization and its conception of star products , and we illustrate the relevance of this point using various examples of dynamical variables quantized via Weyl quantization and coherent state quantization. Our observations suggest that the relevance of the Kochen–Specker theorem to the question of whether one can assign sharp values to all dynamical variables is rather limited.
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.
The Kochen-Specker theorem proves the inability to assign, simultaneously, noncontextual definite values to all (of a finite set of) quantum mechanical observables in a consistent manner. If one assumes that any definite values behave noncontextually, one can nonetheless only conclude that some observables (in this set) are value indefinite. In this paper, we prove a variant of the Kochen-Specker theorem showing that, under the same assumption of noncontextuality, if a single one-dimensional projection observable is assigned the definite value 1, then no one-dimensional projection observable that is incompatible (i.e., non-commuting) with this one can be assigned consistently a definite value. Unlike standard proofs of the Kochen-Specker theorem, in order to localise and show the extent of value indefiniteness, this result requires a constructive method of reduction between Kochen-Specker sets. If a system is prepared in a pure state ψ, then it is reasonable to assume that any value assignment (i.e., hidden variable model) for this system assigns the value 1 to the observable projecting onto the one-dimensional linear subspace spanned by ψ, and the value 0 to those projecting onto linear subspaces orthogonal to it. Our result can be interpreted, under this assumption, as showing that the outcome of a measurement of any other incompatible one-dimensional projection observable cannot be determined in advance, thus formalising a notion of quantum randomness.
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.
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.
The Kochen-Specker theorem, Bell inequalities, and several other tests that were designed to rule out hidden-variable theories, assume the existence of observables having infinitely sharp eigenvalues. A paradigmatic example is spin-1/2. It is measured with a Stern-Gerlach array whose outputs are divided into two classes, spin-up and spin-down, in correspondence to the two spots observed on a detection screen. The spot's finite size is attributed to imperfections of the measuring device. This assumption turns the experimental output into a dichotomic, discrete one, thereby allowing the assignment of each spot to an infinitely sharp eigenvalue. Alternatively, one can assume that the spot's finite size stems from eigenvalues spanning a continuous range. Can we disprove such an assumption? Can we rule out hidden-variable theories that reproduce quantum predictions by assuming that, e.g., the electron's magnetic moment is not exactly the same for all electrons? We address these questions by focusing on the Peres-Mermin version of the Bell-Kochen-Specker theorem. It is shown that the assumption of unsharp eigenvalues precludes ruling out non-contextual hidden-variable theories and hence quantum contextuality does not arise. Analogous results hold for Bell-like inequalities. This represents a new loophole that spoils several fundamental tests of quantum mechanics and issues the challenge to close it.