Contextuality in the Fibration Approach and the Role of 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.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.

arXiv:2105.14132May 28, 20212 citations

Simplicial quantum contextuality

Cihan Okay, Aziz Kharoof, Selman Ipek

We introduce a new framework for contextuality based on simplicial sets, combinatorial models of topological spaces that play a prominent role in modern homotopy theory. Our approach extends measurement scenarios to consist of spaces (rather than sets) of measurements and outcomes, and thereby generalizes nonsignaling distributions to simplicial distributions, which are distributions on spaces modeled by simplicial sets. Using this formalism we present a topologically inspired new proof of Fine's theorem for characterizing noncontextuality in Bell scenarios. Strong contextuality is generalized suitably for simplicial distributions, allowing us to define cohomological witnesses that extend the earlier topological constructions restricted to algebraic relations among quantum observables to the level of probability distributions. Foundational theorems of quantum theory such as the Gleason's theorem and Kochen--Specker theorem can be expressed naturally within this new language.We introduce a new framework for contextuality based on simplicial sets, combinatorial models of topological spaces that play a prominent role in modern homotopy theory. Our approach extends measurement scenarios to consist of spaces (rather than sets) of measurements and outcomes, and thereby generalizes nonsignaling distributions to simplicial distributions, which are distributions on spaces modeled by simplicial sets. Using this formalism we present a topologically inspired new proof of Fine's theorem for characterizing noncontextuality in Bell scenarios. Strong contextuality is generalized suitably for simplicial distributions, allowing us to define cohomological witnesses that extend the earlier topological constructions restricted to algebraic relations among quantum observables to the level of probability distributions. Foundational theorems of quantum theory such as the Gleason's theorem and Kochen--Specker theorem can be expressed naturally within this new language.

arXiv:2204.06648QuantumMay 22, 202310 citations

The Gauge Theoretic Structure of Non-Locality and Contextuality

Imtiyaz, Sahil, Serafim Rodrigues

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.

Jan 1, 20250 citations

Topology of Empirical Models

Imtiyaz, S., Rodrigues, S.

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.

Jan 14, 20250 citations

The geometry of simplicial distributions on suspension scenarios

Aziz Kharoof

Quantum measurements often exhibit non-classical features, such as contextuality, which generalizes Bell's non-locality and serves as a resource in various quantum computation models. Existing frameworks have rigorously captured these phenomena, and recently, simplicial distributions have been introduced to deepen this understanding. The geometrical structure of simplicial distributions can be seen as a resource for applications in quantum information theory. In this work, we use topological foundations to study this geometrical structure, leveraging the fact that, in this simplicial framework, measurements and outcomes are represented as spaces. This allows us to depict contextuality as a topological phenomenon. We show that applying the cone construction to the measurement space makes the corresponding non-signaling polytope equal to the join of $m$ copies of the original polytope, where $m$ is the number of possible outcomes per measurement. Then we glue two copies of cone measurement spaces to obtain a suspension measurement space. The decomposition done for simplicial distributions on a cone measurement space provides deeper insights into the geometry of simplicial distributions on a suspension measurement space and aids in characterizing the contextuality there. Additionally, we apply these results to derive a new type of Bell inequalities (inequalities that determine the set of local joint probabilities/non-contextual simplicial distributions) and to offer a mathematical explanation for certain contextual vertices from the literature.Quantum measurements often exhibit non-classical features, such as contextuality, which generalizes Bell's non-locality and serves as a resource in various quantum computation models. Existing frameworks have rigorously captured these phenomena, and recently, simplicial distributions have been introduced to deepen this understanding. The geometrical structure of simplicial distributions can be seen as a resource for applications in quantum information theory. In this work, we use topological foundations to study this geometrical structure, leveraging the fact that, in this simplicial framework, measurements and outcomes are represented as spaces. This allows us to depict contextuality as a topological phenomenon. We show that applying the cone construction to the measurement space makes the corresponding non-signaling polytope equal to the join of $m$ copies of the original polytope, where $m$ is the number of possible outcomes per measurement. Then we glue two copies of cone measurement spaces to obtain a suspension measurement space. The decomposition done for simplicial distributions on a cone measurement space provides deeper insights into the geometry of simplicial distributions on a suspension measurement space and aids in characterizing the contextuality there. Additionally, we apply these results to derive a new type of Bell inequalities (inequalities that determine the set of local joint probabilities/non-contextual simplicial distributions) and to offer a mathematical explanation for certain contextual vertices from the literature.

arXiv:2412.10963Dec 14, 20240 citations

Logical and topological contextuality in quantum mechanics and beyond

Giovanni Carù

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 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.

Jan 1, 20193 citations

m

copies of the original polytope, where

m

is the number of possible outcomes per measurement. Then we glue two copies of cone measurement spaces to obtain a suspension measurement space. The decomposition done for simplicial distributions on a cone measurement space provides deeper insights into the geometry of simplicial distributions on a suspension measurement space and aids in characterizing the contextuality there. Additionally, we apply these results to derive a new type of Bell inequalities (inequalities that determine the set of local joint probabilities/non-contextual simplicial distributions) and to offer a mathematical explanation for certain contextual vertices from the literature.