Presheaf models provide a formulation of labelled transition systems that is useful for, among other things, modelling concurrent computation. This paper aims to extend such models further to represent stochastic dynamics such as shown in quantum systems. After reviewing what presheaf models represent and what certain operations on them mean in terms of notions such as internal and external choices, composition of systems, and so on, I will show how to extend those models and ideas by combining them with ideas from other category-theoretic approaches to relational models and to stochastic processes. It turns out that my extension yields a transitional formulation of sheaf-theoretic structures that Abramsky and Brandenburger proposed to characterize non-locality and contextuality. An alternative characterization of contextuality will then be given in terms of a dynamic modal logic of the models I put forward.
Presheaf models provide a formulation of labelled transition systems that is useful for, among other things, modelling concurrent computation. This paper aims to extend such models further to represent stochastic dynamics such as shown in quantum systems. After reviewing what presheaf models represent and what certain operations on them mean in terms of notions such as internal and external choices, composition of systems, and so on, I will show how to extend those models and ideas by combining them with ideas from other category-theoretic approaches to relational models and to stochastic processes. It turns out that my extension yields a transitional formulation of sheaf-theoretic structures that Abramsky and Brandenburger proposed to characterize non-locality and contextuality. An alternative characterization of contextuality will then be given in terms of a dynamic modal logic of the models I put forward.
We provide a unified operational framework for the study of causality, non-locality and contextuality, in a fully device-independent and theory-independent setting. Our work has its roots in the sheaf-theoretic framework for contextuality by Abramsky and Brandenburger, which it extends to include arbitrary causal orders (be they definite, dynamical or indefinite). We define a notion of causal function for arbitrary spaces of input histories, and we show that the explicit imposition of causal constraints on joint outputs is equivalent to the free assignment of local outputs to the tip events of input histories. We prove factorisation results for causal functions over parallel, sequential, and conditional sequential compositions of the underlying spaces. We prove that causality is equivalent to continuity with respect to the lowerset topology on the underlying spaces, and we show that partial causal functions defined on open sub-spaces can be bundled into a presheaf. In a striking departure from the Abramsky-Brandenburger setting, however, we show that causal functions fail, under certain circumstances, to form a sheaf. We define empirical models as compatible families in the presheaf of probability distributions on causal functions, for arbitrary open covers of the underlying space of input histories. We show the existence of causally-induced contextuality, a phenomenon arising when the causal constraints themselves become context-dependent, and we prove a no-go result for non-locality on total orders, both static and dynamical.
This thesis presents two new approaches to two topics in quantum foundations: Part I: The study of causal structures and higher order processes, i.e., transformations of processes.Part II: The study of non-locality and contextuality using effect algebras.embed in a category of presheaves and this allows us to make a connection between the two approaches using adjunctions.In particular, we are able to relate our effect algebraic framework to the established presheaf approach of Abramsky and Brandenburger.In a similar fashion we relate effect algebras to test spaces.We describe explicitly the examples of Bell, Kochen-Specker, Hardy and GHZ.
The study of the foundations of Quantum Mechanics, especially after the advent of Quantum Computation and Information, has benefited from the application of category-theoretic tools and modal logics to the analysis of Quantum processes: we witness a wealth of theoretical frameworks casted in either of the two languages. This paper explores the interplay of the two formalisms in the peculiar context of Quantum Theory. After a review of some influential abstract frameworks, we show how different modal logic frames can be extracted from the category of finite dimensional Hilbert spaces, connecting the Categorical Quantum Mechanics approach to some modal logics that have been proposed for Quantum Computing. We then apply a general version of the same technique to two other categorical frameworks, the ‘topos approach’ of Doering and Isham and the sheaf-theoretic work on contextuality by Abramsky and Brandenburger, suggesting how some key features can be expressed with modal languages.
Extendability of an empirical model was shown by Abramsky & Brandenburger to correspond in a unified manner to both locality and non-contextuality. We develop their approach by presenting a refinement of the notion of extendability that can also be useful in characterising the properties of sub-models. The refinement is found to have another useful application: it is shown that a particular canonical extension, when well-defined, may be used for the construction of Bell-type models from models of more general kinds in such a way that the constructed model is equivalent to the original in terms of non-locality/contextuality. This is important since on practical and foundational levels, the notion of locality in Bell-type models can more easily be motivated than the corresponding general notion of contextuality. We consider examples of Bell-type models generated from some standard examples of contextual models, including an entire class of Kochen-Specker-like models. This exposes an intriguing relationship between the simplest possible contextual model (the contextual triangle) and Popescu-Rohrlich no-signalling correlations.
We develop an extended formulation of Relational Quantum Dynamics (RQD) in which the quantum wave function is treated as an intrinsic relational entity rather than an external absolute state. Building on enriched category theory and related advanced frameworks, we formalize RQD using higher-category structures and presheaf/topos-theoretic tools to rigorously capture the web of relations between quantum systems (observers and observed). In this enriched formalism, each pair of quantum systems is associated with a relational Hilbert space and state vector, and higher-category morphisms naturally represent multi-observer interactions. We incorporate information-theoretic “awareness” measures such as mutual information and integrated information into the theory, allowing a quantitative gauge of relational knowledge or consciousness where applicable. We show that the Born rule for outcome probabilities emerges internally from the relational structure, consistent with Gleason’s theorem, and that quantum contextuality (no observer-independent values) is built in by construction. Using presheaf models, we demonstrate that no global hidden-variable state can exist across all observers, aligning with the Kochen–Specker theorem. The extended RQD remains empirically equivalent to standard quantum mechanics in ordinary scenarios, but it provides a richer language to address exotic setups. We explore novel experimental scenarios, including designs where quantum computers serve as controllable “observers.” We review recent work in quantum computation and relational interpretations, and propose how multi-observer interference experiments and quantum information protocols can test the relational framework.
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.