Quantum measurement is commonly posed as a dynamical tension between linear Schrödinger evolution and an ad hoc collapse rule. I argue that the deeper conflict is logical: quantum theory is inherently contextual, whereas the classical tradition presupposes a single global, Boolean valuation. Building on Bohr's complementarity, the Einstein--Podolsky--Rosen argument and Bell's theorem, I recast locality and completeness as the existence of a global section of a presheaf of value assignments over the category of measurement contexts. The absence of global sections expresses the impossibility of context-independent description, and Čech cohomology measures the resulting obstruction. The internal logic of the presheaf topos is intuitionistic, and the seven-valued contextual logic proposed by Ghose and Patra is exhibited as a finite Heyting algebra capturing patterns of truth, falsity and indeterminacy across incompatible contexts. Classical physics corresponds to the sheaf case, where compatible local data glue and Boolean logic is effectively restored. Measurement is therefore reinterpreted as sheafification of presheaf-valued truth rather than as a physical breakdown of unitarity. Finally, a $σ$--$λ$ dynamics motivated by stochastic mechanics provides a continuous interpolation between strongly contextual and approximately classical regimes, dissolving the usual measurement paradoxes and apparent nonlocality as artefacts of an illegitimate demand for global truth.
Comments
23 pages, no figures; a short section om cohomology added to Appendix A for completeness
Quantum measurement is commonly posed as a dynamical tension between linear Schrödinger evolution and an ad hoc collapse rule. I argue that the deeper conflict is logical: quantum theory is inherently contextual, whereas the classical tradition presupposes a single global, Boolean valuation. Building on Bohr's complementarity, the Einstein--Podolsky--Rosen argument and Bell's theorem, I recast locality and completeness as the existence of a global section of a presheaf of value assignments over the category of measurement contexts. The absence of global sections expresses the impossibility of context-independent description, and Čech cohomology measures the resulting obstruction. The internal logic of the presheaf topos is intuitionistic, and the seven-valued contextual logic proposed by Ghose and Patra is exhibited as a finite Heyting algebra capturing patterns of truth, falsity and indeterminacy across incompatible contexts. Classical physics corresponds to the sheaf case, where compatible local data glue and Boolean logic is effectively restored. Measurement is therefore reinterpreted as sheafification of presheaf-valued truth rather than as a physical breakdown of unitarity. Finally, a σ--λ dynamics motivated by stochastic mechanics provides a continuous interpolation between strongly contextual and approximately classical regimes, dissolving the usual measurement paradoxes and apparent nonlocality as artefacts of an illegitimate demand for global truth.
The main subjects of this thesis are non-locality and contextuality, two fundamental features of quantum mechanics that constitute valuable resources for quantum computation. Our analysis is based on Abramsky & Brandenburger's sheaf theoretic framework, which captures both these phenomena in a unified treatment and in a very general setting. This high-level description transcends quantum physics and allows to precisely characterise the notion of contextuality as the apparent paradox realised by data being locally consistent but globally inconsistent. More specifically, we aim to develop a deeper understanding of so-called logical forms of contextuality, i.e. situations where the phenomenon can be witnessed using purely logical arguments, disregarding probabilities. The sheaf theoretic description of logical contextuality has recently inspired the development of a topological treatment of the phenomenon based on sheaf cohomology. In this thesis, we embark on a detailed analysis of the cohomology of contextuality, exposing key shortcomings in the current methods, and introducing an (almost) complete cohomological characterisation of logical forms of contextuality. More specifically, we show that, in its current formulation, sheaf cohomology does not constitute a complete invariant for contextuality, not even in its strongest forms, and that higher cohomology groups cannot be used to study the phenomenon. Then, we solve these issues by introducing a novel construction, which derives refined versions of the presheaves describing empirical models to expose their deeper extendability properties, resulting in a sheaf cohomological invariant which is applicable to the vast majority of empirical models, and conjectured to work universally. We propose a general theory of contextual semantics using the language of valuation algebras. In particular, we give a general definition of contextual behaviour as a fundamental gap between local agreement and global disagreement of information sources. Not only does this formalism aptly capture and generalise the known instances of contextuality beyond quantum theory, but it also provides inspiration for further applications of the phenomenon, and paves the way for the transfer of results and techniques between different fields. We give a prime example of this potential by developing faster algorithms to detect contextuality based on mainstream methods of generic inference. Finally, we turn our attention back to instances of contextuality in quantum physics, and study strong contextuality in multi-qubit states. We give a complete combinatorial characterisation of All-vs-Nothing proofs of strong contextuality in stabiliser quantum mechanics. This allows to produce the complete list of all stabiliser states exhibiting this kind of contextuality, which consitututes an important resource in certain models of quantum computation. Then, we extend our search for strongly contextual behaviour beyond stabiliser states, and identify the minimum quantum resources needed to realise strong non-locality. Additional results include a partial classification of strongly non-local models comprised of three-qubit states and local projective measurements, and the introduction of a new infinite family of strongly non-local three-qubit states.
The semantics underlying the propositional structure of Hilbert space quantum mechanics involves an inherent ambiguity concerning the impossibility of assigning definite truth values to all propositions pertaining to a quantum system without generating a Kochen-Specker contradiction. Although the preceding Kochen-Specker result forbids a global, absolute assignment of truth values to quantum mechanical propositions, it does not exclude ones that are contextual. In this respect, the Bub-Clifton “uniqueness theorem” is utilized for arguing that truth-value definiteness is consistently restored with respect to a determinate sublattice of propositions defined by the state of the quantum system concerned and a suitable “preferred” observable. It is suggested that the most natural choice of the latter, especially for confronting the problem of truth valuation in quantum mechanics, results in the determinateness of the observable to be measured. An account of truth of contextual correspondence is thereby provided that is appropriate to the quantum domain of discourse. The resulting account of truth is compatible with a realist conception of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature.
In this thesis we use the language of sheaf theory in order to develop a deeper understanding of some of the fundamental differences - such as entanglement, contextuality and non-locality - between quantum and classical physics. We first present, based on the work of Abramsky and Brandenburger, how sheaves, defined over certain posets of physically meaningful contexts, give a natural setting for capturing and analysing important quantum mechanical phenomena, such as quantum non-locality and contextuality. We also describe how this setting naturally leads to a three level hierarchy of quantum contextuality: weak contextuality, logical non-locality and strong contextuality. We use these insights in order to classify certain multipartite entangled states. Almost all of these turn out to be at least logically non-local, and a number of them even turn out to be strongly contextual. We further extend this result by showing that all n-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, are logically non-local. Our proof is constructive: given any n-qubit state, we present an algorithm which produces n+2 local observables witnessing its logical non-locality. In the second half of the thesis we use the same basic principle of sheaves defined over physically meaningful contexts, in order to present an elegant mathematical language, known under the name of the Topos Approach, in which many quantum mechanical concepts, such as states, observables, and propositions about these, can be expressed. We then show that the language of the Topos Approach is as least as expressive, in logical terms, as traditional quantum logic. Finally, starting from a topos-theoretic perspective, we develop the construction of contextual entropy in order to give a unified treatment of classical and quantum notions of information theoretic entropy.
This paper argues that physical quantities should be reconstructed not as non-contextual intrinsic attributes with measurement-independent determinate values, but as equivalence classes of measurement procedures. Bell–CHSH violations first motivate the problem: they show that Bell-type local causality/factorizability, measurement independence, and the relevant form of outcome determinism or value-definiteness cannot be jointly maintained, without by themselves refuting realism as such. Kochen–Specker contextuality supplies the stronger premise for the sharp-observable case: no global, non-contextual, dispersion-free value assignment to projection-valued observables can preserve the functional relations among compatible observables in Hilbert spaces of dimension at least three. The positive proposal uses algebraic quantum theory to identify physical quantities with classes of measuring procedures that yield the same outcome statistics over a full or tomographically complete state domain. This reconstruction preserves the classical deterministic limit while rejecting the stronger claim that measured quantities are context-independent values possessed independently of measurement. The analysis is restricted to sharp observables and projection-valued measurements; POVMs and unsharp measurements are noted as extensions. The resulting view is realist about an observer-independent world but non-intrinsic about empirical quantities, and it situates Wheeler's "It from Bit" as an interpretive corollary rather than as a foundational premise.
The contextual character of quantum observables—formalized by the Kochen--Specker theorem and Bohr's complementarity—represents a fundamental rupture in classical binary logic. In interference regimes, such as the double-slit experiment, polar questions (e.g., ``Which path did the particle take?'') are operationally unaskable, yet standard logical frameworks fail to formalize this unaskability without collapsing into truth-value gaps or triviality. We resolve this by introducing a context-indexed formal logic that adjoins an explicit \emph{applicability predicate} to the valuation space. Drawing structural inspiration from the ancient \emph{catuṣkoṭi} (tetralemma), we map quantum operational states onto four distinct logical sectors: \emph{Thesis/Antithesis} (strong measurement and decoherence), \emph{Synthesis} (paraconsistent gluts realized via weak measurement), and \emph{Holothesis} (paracomplete gaps representing typed inapplicability). From this logical foundation, we generate a ∗-algebra equipped with a positive warrant functional. Through the Gelfand--Naimark--Segal (GNS) construction, we demonstrate that Hilbert-space structure—and the complex unit i itself—emerge canonically from an order-4 logical symmetry rather than being presupposed. Contextuality is quantified via incompatibility graphs, with chromatic number bounding minimal classical embeddings. We establish a Noether-like correspondence wherein physical symmetry breaking dictates logical polarity, while symmetry preservation dictates logical transcendence (self-duality). This \emph{logic-first reconstruction} unifies Relational Quantum Mechanics (Rovelli) and Consistent Histories (Griffiths, Omnès, Gell-Mann, Hartle) as regime-relative logical stances within a pre-geometric ontology of \emph{interdependent origination} (contextual co-arising of observer and observable), where the observer's contextual frame and the system's observable reality are co-arising and non-separable. The framework yields testable deviations in many-body interference (I3=0), providing an empirical pathway for detection.
The contextual character of quantum observables—formalized by the Kochen--Specker theorem and Bohr's complementarity—represents a fundamental rupture in classical binary logic. In interference regimes, such as the double-slit experiment, polar questions (e.g., ``Which path did the particle take?'') are operationally unaskable, yet standard logical frameworks fail to formalize this unaskability without collapsing into truth-value gaps or triviality. We resolve this by introducing a context-indexed formal logic that adjoins an explicit \emph{applicability predicate} to the valuation space. Drawing structural inspiration from the ancient \emph{catuṣkoṭi} (tetralemma), we map quantum operational states onto four distinct logical sectors: \emph{Thesis/Antithesis} (strong measurement and decoherence), \emph{Synthesis} (paraconsistent gluts realized via weak measurement), and \emph{Holothesis} (paracomplete gaps representing typed inapplicability). From this logical foundation, we generate a ∗-algebra equipped with a positive warrant functional. Through the Gelfand--Naimark--Segal (GNS) construction, we demonstrate that Hilbert-space structure—and the complex unit i itself—emerge canonically from an order-4 logical symmetry rather than being presupposed. Contextuality is quantified via incompatibility graphs, with chromatic number bounding minimal classical embeddings. We establish a Noether-like correspondence wherein physical symmetry breaking dictates logical polarity, while symmetry preservation dictates logical transcendence (self-duality). This \emph{logic-first reconstruction} unifies Relational Quantum Mechanics (Rovelli) and Consistent Histories (Griffiths, Omnès, Gell-Mann, Hartle) as regime-relative logical stances within a pre-geometric ontology of \emph{interdependent origination} (contextual co-arising of observer and observable), where the observer's contextual frame and the system's observable reality are co-arising and non-separable. The framework yields testable deviations in many-body interference (I3=0), providing an empirical pathway for detection.