The Sheaf-Theoretic Structure Of Non-Locality and Contextuality

Abstract

We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, in a setting which generalizes the familiar probability tables used in non-locality theory to arbitrary measurement covers; this includes Kochen-Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections. We describe a linear algebraic approach to computing these obstructions, which allows a systematic treatment of arguments for non-locality and contextuality. We distinguish a proper hierarchy of strengths of no-go theorems, and show that three leading examples --- due to Bell, Hardy, and Greenberger, Horne and Zeilinger, respectively --- occupy successively higher levels of this hierarchy. A general correspondence is shown between the existence of local hidden-variable realizations using negative probabilities, and no-signalling; this is based on a result showing that the linear subspaces generated by the non-contextual and no-signalling models, over an arbitrary measurement cover, coincide. Maximal non-locality is generalized to maximal contextuality, and characterized in purely qualitative terms, as the non-existence of global sections in the support. A general setting is developed for Kochen-Specker type results, as generic, model-independent proofs of maximal contextuality, and a new combinatorial condition is given, which generalizes the `parity proofs' commonly found in the literature. We also show how our abstract setting can be represented in quantum mechanics. This leads to a strengthening of the usual no-signalling theorem, which shows that quantum mechanics obeys no-signalling for arbitrary families of commuting observables, not just those represented on different factors of a tensor product.

33 pages. Extensively revised, new results included. Published in New Journal of Physics

References in corpus (4)

• State-independent experimental test of quantum contextuality
• Experimental test of quantum contextuality in neutron interferometry
• A topos for algebraic quantum theory
• A Classification of Hidden-Variable Properties

Cited by in corpus (227)

• Almost quantum correlations
• Kochen-Specker Contextuality
• A Combinatorial Approach to Nonlocality and Contextuality
• All noncontextuality inequalities for the n-cycle scenario
• The contextual fraction as a measure of contextuality
• Wigner function negativity and contextuality in quantum computation on rebits
• Logical Bell Inequalities
• The Inflation Technique for Causal Inference with Latent Variables
• Contextuality and Wigner function negativity in qubit quantum computation
• An entropic approach to local realism and noncontextuality
• Entropic Inequalities and Marginal Problems
• True Contextuality Beats Direct Influences in Human Decision Making
• Entropic Test of Quantum Contextuality
• A free energy principle for generic quantum systems
• The Cohomology of Non-Locality and Contextuality
• Causal Compatibility Inequalities Admitting Quantum Violations in the Triangle Structure
• Equivalence between contextuality and negativity of the Wigner function for qudits
• Phase space simulation method for quantum computation with magic states on qubits
• Contextuality is About Identity of Random Variables
• Contextuality in Canonical Systems of Random Variables
• Machine Learning meets Quantum Foundations: A Brief Survey
• Laws of Nature as Constraints
• Quantum realization of arbitrary joint measurability structures
• Quantum Advantage from Sequential-Transformation Contextuality
• Information complementarity: A new paradigm for decoding quantum incompatibility
• Properties and Applications of the Kirkwood-Dirac Distribution
• Contextuality Test of the Nonclassicality of Variational Quantum Eigensolvers
• Contextuality as a resource for measurement-based quantum computation beyond qubits
• Noncontextual wirings
• Resource theory of contextuality for arbitrary prepare-and-measure experiments
• Contextuality and Wigner negativity are equivalent for continuous-variable quantum measurements
• Optimal classical simulation of state-independent quantum contextuality
• Measures of Contextuality and Noncontextuality
• Interference of Quantum Trajectories
• Simulating all non-signalling correlations via classical or quantum theory with negative probabilities
• Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces
• Categorical Probabilistic Theories
• Two Proofs of Fine's Theorem
• One-Shot Manipulation of Dynamical Quantum Resources
• Unified approach to contextuality, non-locality, and temporal correlations
• Classical causal models for Bell and Kochen-Specker inequality violations require fine-tuning
• Entropic nonsignaling correlations
• Contextuality of general probabilistic theories
• Bounding quantum contextuality with lack of third-order intereference
• Quantum Team Logic and Bell's Inequalities
• Quantum correlations from simple assumptions
• Proof of the Peres conjecture for contextuality
• Contextual Subspace Variational Quantum Eigensolver
• Experimental test of contextuality in quantum and classical systems
• Quantum Correlations and Quantum Non-locality: a review and a few new ideas
• Analysing causal structures with entropy
• The Free Energy Principle drives neuromorphic development
• Experimental Demonstration of Quantum Pseudotelepathy
• A possibilistic no-go theorem on the Wigner's friend paradox
• Measuring incompatibility and clustering quantum observables with a quantum switch
• Probing the limits of correlations in an indivisible quantum system
• Reality of the quantum state: Towards a stronger ψ-ontology theorem
• Quantum computational advantage with string order parameters of 1D symmetry-protected topological order
• Quantum Advantage in Information Retrieval
• Contextuality-by-Default description of Bell tests: Contextuality as the rule not as an exception
• Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements
• Almost Quantum Correlations are Inconsistent with Specker's Principle
• Classical Simulation of Noncontextual Pauli Hamiltonians
• Inequalities witnessing coherence, nonlocality, and contextuality
• Contextual inferences, nonlocality, and the incompleteness of quantum mechanics
• Geometry of contextuality from Grothendieck's coset space
• Macroscopic non-contextuality as a principle for Almost Quantum Correlations
• Quantum theory realises all joint measurability graphs
• Graph-theoretic strengths of contextuality
• Non-deterministic semantics for quantum states
• Neither Contextuality nor Nonlocality Admits Catalysts
• A comonadic view of simulation and quantum resources
• On the Cohomology of Contextuality
• Fine's Theorem for Leggett-Garg tests with an arbitrary number of measurement times
• The complete Heyting algebra of subsystems and contextuality
• Fine's theorem, noncontextuality, and correlations in Specker's scenario
• Cellular Sheaves of Lattices and the Tarski Laplacian
• Team Decision Problems with Classical and Quantum Signals
• The role of cohomology in quantum computation with magic states
• Classical causal models cannot faithfully explain Bell nonlocality or Kochen-Specker contextuality in arbitrary scenarios
• On Joint Distributions, Counterfactual Values, and Hidden Variables in Relation to Contextuality
• A mathematical framework for operational fine tunings
• Necessary Conditions for Extended Noncontextuality in General Sets of Random Variables
• Violation of Bell inequalities in an analogue black hole
• A proof of the Kochen-Specker theorem can always be converted to a state-independent noncontextuality inequality
• The many classical faces of quantum structures
• A classification of multipartite states by degree of non-locality
• Non-locality, contextuality and valuation algebras: a general theory of disagreement
• Contextuality as a Precondition for Quantum Entanglement
• A Categorial Semantic Representation of Quantum Event Structures
• Assignments to sheaves of pseudometric spaces
• Consequences and applications of the completeness of Hardy's nonlocality
• Replacing Nothing with Something Special: Contextuality-by-Default and Dummy Measurements
• Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation
• Conditions for Macrorealism for Systems Described by Many-Valued Variables
• Separability, Contextuality, and the Quantum Frame Problem
• Quantum Theory is a Quasi-stochastic Process Theory
• On Measures and Measurements: a Fibre Bundle approach to Contextuality
• Homotopical approach to quantum contextuality
• Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem
• Contextuality in multipartite pseudo-telepathy graph games
• Hardy is (almost) everywhere: nonlocality without inequalities for almost all entangled multipartite states
• Graph Approach to Extended Contextuality
• Reliable computation from contextual correlations
• No-Signalling Is Equivalent To Free Choice of Measurements
• On monogamy of non-locality and macroscopic averages: examples and preliminary results
• Frobenius structures over Hilbert C*-modules
• Contextuality, Fine-Tuning, and Teleological Explanation
• Contextuality degree of quadrics in multi-qubit symplectic polar spaces
• Dual form of the phase-space classical simulation problem in quantum optics
• Observers of quantum systems cannot agree to disagree
• Contextuality in composite systems: the role of entanglement in the Kochen-Specker theorem
• Hierarchies of resources for measurement-based quantum computation
• Simplicial quantum contextuality
• A representation of the wave function on the three-dimensional space
• Contextual bound states for qudit magic state distillation
• Contextuality and bundle diagrams
• Certified randomness in tight space
• A complete characterisation of All-versus-Nothing arguments for stabiliser states
• Universal resources for quantum computing
• Categories of Empirical Models
• Possibilities Determine the Combinatorial Structure of Probability Polytopes
• On Possibilistic Conditions to Contextuality and Nonlocality
• Generalizing optimal Bell inequalities
• The power of qutrits for non-adaptive measurement-based quantum computing
• Contextuality under weak assumptions
• Cohomology of Effect Algebras
• Contextuality and noncommutative geometry in quantum mechanics
• Assumption-free derivation of the Bell-type criteria of contextuality/nonlocality
• The Sheaf-Theoretic Structure of Definite Causality
• Corrected Bell and Noncontextuality Inequalities for Realistic Experiments
• Logical and inequality based contextuality for qudits
• A Bestiary of Sets and Relations
• On the tightness of correlation inequalities with no quantum violation
• The rank of contextuality
• Nonlocality with less Complementarity
• Closing Bell: Boxing black box simulations in the resource theory of contextuality
• Entropic No-Disturbance as a Physical Principle
• Playing nonlocal games with phases of quantum matter
• Contexts in Convex and Sequential Effect Algebras
• Cohomology and the Algebraic Structure of Contextuality in Measurement Based Quantum Computation
• Physics is organized around transformations connecting contextures in a polycontextural world
• Properties of 2D anyon gas
• A multi-player, multi-team nonlocal game for the toric code
• Rényi Entropy, Signed Probabilities, and the Qubit
• Bell non-locality using tensor networks and sparse recovery
• Continuous-variable nonlocality and contextuality
• A Combinatorial Solution to Causal Compatibility
• On the origin of nonclassicality in single systems
• Deriving the Qubit from Entropy Principles
• Topological methods for studying contextuality: $N$-cycle scenarios and beyond
• Almost Equivalent Paradigms of Contextuality
• Violating the KCBS inequality with a toy mechanism
• Modelling contextuality by probabilistic programs with hypergraph semantics
• Connecting XOR and XOR* games
• Freudenthal ranks: GHZ vs. W
• Logical Entropy and Negative Probabilities in Quantum Mechanics
• The problem of quantum correlations and the totalitarian principle
• Potential unsatisfiability of cyclic constraints on stochastic biological networks biases selection toward hierarchical architectures
• A Logical Proof of Quantum Correlations Requiring Entanglement Measurements
• Nonclassical correlations of radiation in relation to Bell nonlocality
• Combining contextuality and causality: a game semantics approach
• Mermin Non-Locality in Abstract Process Theories
• Tsirelson inequalities: Detecting cheating and quantumness in a single framework
• Extended Wigner's friend paradoxes do not require nonlocal correlations
• Consistent circuits for indefinite causal order
• Higher-order CPM Constructions
• A Model of Anaphoric Ambiguities using Sheaf Theoretic Quantum-like Contextuality and BERT
• Lifting noncontextuality inequalities
• Classifying space for quantum contextuality
• Witnessing Bell violations through probabilistic negativity
• Local Friendliness Polytopes In Multipartite Scenarios
• Multipartite Composition of Contextuality Scenarios
• Wigner and friends, a map is not the territory! Contextuality in multi-agent paradoxes
• Reflections on the PBR Theorem: Reality Criteria & Preparation Independence
• Exploring the boundary of quantum correlations with a time-domain optical processor
• Topos logic in measurement-based quantum computation
• Reexamination of the Kochen-Specker theorem: Relaxation of the completeness assumption
• Stochastic Relational Presheaves and Dynamic Logic for Contextuality
• Generalised Winograd Schema and its Contextuality
• Contextuality and Informational Redundancy
• Analysis of single-particle nonlocality through the prism of weak measurements
• Contextuality with disturbance and without: Neither can violate substantive requirements the other satisfies
• The six blinds and the elephant or an interdisciplinary selection of measurement features
• Newton's laws of motion generating gravity-mediated entanglement
• Connecting extended Wigner's friend arguments and noncontextuality
• Complete extension: the non-signaling analog of quantum purification
• Contextuality Can be Verified with Noncontextual Experiments
• Sufficiency of the counterfactual account of Lüders' rule to rule out ontological models of quantum mechanics
• Deterministic Underlying States are incompatible with a Counterfactual account of Lüders' rule
• Contextuality of Quantum Error-Correcting Codes
• Harvesting Contextuality from the Vacuum
• Contemporary Perspectivism as a Framework of Scientific Inquiry in Quantum Mechanics and Beyond
• Contextuality scenarios arising from networks of stochastic processes
• Twenty years of quantum contextuality at USTC
• Physical properties as modal operators in the topos approach to quantum mechanics
• An Entanglement Monotone from the Contextual Fraction
• Contextuality based quantum conferencing
• On Information Links
• Maximal Non-Kochen-Specker Sets and a Lower Bound on the Size of Kochen-Specker Sets
• Contextuality and dichotomizations of random variables
• Atom graph, partial Boolean algebra and quantum contextuality
• Classical Invasive Description of Informationally-Complete Quantum Processes
• Experimental Approach to Demonstrating Contextuality for Qudits
• Contextuality and Expressivity of Non-locality
• All fundamental non-contextuality inequalities are unique
• Differential Geometry of Contextuality
• The Causal Structure of Semantic Ambiguities
• Hidden variables, free choice, context-independence, and all that
• Contextuality, superlocality and nonclassicality of supernoncontextuality
• Quantum HodgeRank: Topology-Based Rank Aggregation on Quantum Computers
• Latent optical nonclassicality of conditionally prepared states
• Noncontextual Pauli Hamiltonians
• Quantum contextuality from measurement invasiveness
• k-Contextuality as a Heuristic for Memory Separations in Learning
• Contextuality from the vacuum
• Symmetry-Accelerated Classical Simulation of Clifford-Dominated Circuits
• Gedanken Experiments of Entanglement in Particle Physics: Interactions, Operators and Bell Inequalities in Flavor Space
• Analogs of absolutely maximally entangled states in nonlocal correlations via the sheaf-theoretic framework and its applications
• Perspectivist Account of Truth-Theoretic Semantics in Quantum Mechanics
• Degrees, Levels, and Profiles of Contextuality
• A Classification Program for Nonlocality Paradoxes of Three Qubits
• Convexity of noncontextual wirings and how they order the set of correlations
• Comparing the cost of violating causal assumptions in Bell experiments: locality, free choice and arrow-of-time
• Quantum mechanics is compatible with counterfactual definiteness
• Violation of no signaling in higher order quantum measure theories
• The Interplay between Quantum Contextuality and Wigner Negativity