Abramsky and Branderburger put forth a sheaf based interpretation of non-locality and contextuality as obstructions in inter-knitting local sections together to form a compatible global section. The synergistic interaction of a local function(s) in a global topological space unfolds novel behaviour emerging as a locally consistent but globally inconsistent pattern in data characterising this empirical phenomenon. We explore a general method to associate an explicit approximation of a topological space represented as a simplicial complex to the empirical models, witness- ing different strengths of contextuality; alongside their polyhedral description whose symmetries are subject to this structural constraint akin to sheafification based on information of tabular representation of these models. A local consistency corresponds to any possible polyhedral symmetry whereas a global consistency characterises symmetries that take a polyhedron back to itself subject to spatial constraints whose discrete representation quantifies contex- tuality through strong collapse as (non)existence of critical simplices and virtual loops using discrete Morse theory. The transient virtual loops characterise contextuality as a topological phase transition – a change in homotopy class – that turns relations locally consistent for an observer but globally non-extendable. We apply the framework on several empirical models in the foundation of quantum physics. The framework could provide a practical way to propose new models for witnessing higher dimensional contextuality in guiding physical experiments and linking the phenomenon to the evolution of geometric structures on 3-manifold theory. We provide two new basic models as examples to conceptualise the reverse of our framework of reproducing possibly higher dimensional tables from a given space and associated polyhedron that could lead to observation of new strength of hyper-contextual scenarios. A seven dimensional Mermin-Ardehali-Belinskii-Klyshko model with its graph-based description could be a first step to discover new structures on 3-manifolds for higher dimensional contextuality.
Abramsky and Branderburger put forth a sheaf based interpretation of non-locality and contextuality as obstructions in inter-knitting local sections together to form a compatible global section. The synergistic interaction of a local function(s) in a global topological space unfolds novel behaviour emerging as a locally consistent but globally inconsistent pattern in data characterising this empirical phenomenon. We explore a general method to associate an explicit approximation of a topological space represented as a simplicial complex to the empirical models, witness- ing different strengths of contextuality; alongside their polyhedral description whose symmetries are subject to this structural constraint akin to sheafification based on information of tabular representation of these models. A local consistency corresponds to any possible polyhedral symmetry whereas a global consistency characterises symmetries that take a polyhedron back to itself subject to spatial constraints whose discrete representation quantifies contex- tuality through strong collapse as (non)existence of critical simplices and virtual loops using discrete Morse theory. The transient virtual loops characterise contextuality as a topological phase transition – a change in homotopy class – that turns relations locally consistent for an observer but globally non-extendable. We apply the framework on several empirical models in the foundation of quantum physics. The framework could provide a practical way to propose new models for witnessing higher dimensional contextuality in guiding physical experiments and linking the phenomenon to the evolution of geometric structures on 3-manifold theory. We provide two new basic models as examples to conceptualise the reverse of our framework of reproducing possibly higher dimensional tables from a given space and associated polyhedron that could lead to observation of new strength of hyper-contextual scenarios. A seven dimensional Mermin-Ardehali-Belinskii-Klyshko model with its graph-based description could be a first step to discover new structures on 3-manifolds for higher dimensional contextuality.
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.
Abramsky and Brandenburger show that nonlocality and contextuality arise when locally compatible data cannot be glued into a single global assignment. In a global topological data space, interactions among locally defined maps can produce patterns that are consistent within each context yet incompatible globally—the empirical hallmark of quantum contextuality. We make this concrete with an evolving simplicial-bundle framework and apply this framework to several standard empirical models in quantum foundations.
Contextuality can be understood as the impossibility to construct a globally consistent description of a model even if there is local agreement. In particular, quantum models present this property. We can describe contextuality with the fibration approach, where the scenario is represented as a simplicial complex, the fibers being the sets of outcomes, and contextuality as the non-existence of a global section in the measure fibration, allowing direct representation and formalization of the already used bundle diagrams. Using the generalization to continuous outcome fibers, we built the concept of measure fibration, showing the Fine-Abramsky-Brandenburger theorem for the fibration formalism in the case of non-finite fibers. By the Voroby'ev theorem, we argue that the dependence of contextual behavior of a model to the topology of the scenario is an open problem. We introduce a hierarchy of contextual behavior to explore it, following the construction of the simplicial complex. GHZ models show that quantum theory has all levels of the hierarchy, and we exemplify the dependence on higher homotopical groups by the tetraedron scenario, where non-trivial topology implies an increase of contextual behavior for this case. For the first level of the hierarchy, we construct the concept of connection through Markov operators for the measure bundle using the measure on fibers of contexts with two measurements and taking the case of equal fibers we can identify the outcome space as the basis of a vector space, that transform according to a group extracted from the connection. With this, it is possible to show that contextuality at the level of contexts with two measurements has a relationship with the non-triviality of the holonomy group in the frame bundle. We give examples and treat disturbing models through transition functions, generalizing the holonomy.
Contextuality can be understood as the impossibility to construct a globally consistent description of a model even if there is local agreement. In particular, quantum models present this property. We can describe contextuality with the bundle approach, where the scenario is represented as a simplicial complex, the fibers being the sets of outcomes, and contextuality as the non-existence of global section in the measure bundle. Using the generalization to non-finite outcome fibers, we built in details the concept of measure bundle, demonstrating the Fine-Abramsky-Brandenburger theorem for the bundle formalism. We introduce a hierarchy called n-contextuality to explore the dependence of contextual behavior of a model to the topology of the scenario, following the construction of it as a simplicial complex. With it we exemplify the dependence on higher homology groups and show that GHZ models, thus quantum theory, has all levels of the hierarchy. Also, we give an example of how non-trivial topology of the scenario result an increase of contextual behavior. For the first level of the hierarchy, we construct the concept of connection through Markov operators for the measure bundle. Taking the case of equal fibers we can identify the outcomes as the basis of a vector space, that transform according to a group extracted from the connection. We thus show that contextuality has a relationship with the non-triviality of the holonomy group in the respective frame bundle.
Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of global sections. In the present work, we illustrate new insights into different aspects of this theory. We shed light on the power of detection of the cohomological obstruction by showing that it is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions on the measurement cover, disproving a previous conjecture. We generalise obstructions to higher cohomology groups and show that they give rise to a refinement of the notion of cohomological contextuality: different "levels" of contextuality are organised in a hierarchy of logical implications. Finally, we present an alternative description of the first cohomology group in terms of torsors, resulting in a new interpretation of the cohomological obstructions.
Contextuality in quantum physics provides a key resource for quantum information and computation. The topological approach in [Abramsky and Brandenburger, New J. Phys., 2011, Abramsky et al., CSL 2015, 2015] characterizes contextuality as "global inconsistency" coupled with "local consistency", revealing it to be a phenomenon also found in many other fields. This has yielded a logical method of detecting and proving the "global inconsistency" part of contextuality. Our goal is to capture the other, "local consistency" part, which requires a novel approach to logic that is sensitive to the topology of contexts. To achieve this, we formulate a logic of local inference by using context-sensitive theories and models in regular categories. This provides a uniform framework for local consistency, and lays a foundation for high-level methods of detecting, proving, and moreover using contextuality as computational resource.