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.
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.
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.
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.
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.
The sheaf theoretic description of non-locality and contextuality by Abramsky and Brandenburger sets the ground for a topological study of these peculiar features of quantum mechanics. This viewpoint has been recently developed thanks to sheaf cohomology, which provides a sufficient condition for contextuality of empirical models in quantum mechanics and beyond. Subsequently, a number of studies proposed methods to detect contextuality based on different cohomology theories. However, none of these cohomological descriptions succeeds in giving a full invariant for contextuality applicable to concrete examples. In the present work, we introduce a cohomology invariant for possibilistic and strong contextuality which is applicable to the vast majority of empirical models.
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.
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.