Simplicial methods in the resource theory of contextuality

• URL: http://arxiv.org/abs/2505.24010v1
• Date: Thu, 29 May 2025 21:14:55 GMT
• Title: Simplicial methods in the resource theory of contextuality
• Authors: Aziz Kharoof, Cihan Okay,
• Abstract summary: Building on the theory of simplicial distributions, we introduce event scenarios as a functorial generalization of presheaf-theoretic measurement scenarios.<n>We define symmetric monoidal structures on these categories and extend the distribution functor to a setting, yielding a resource theory that generalizes the presheaf-theoretic notion of simulations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a resource theory of contextuality within the framework of symmetric monoidal categories, extending recent simplicial approaches to quantum contextuality. Building on the theory of simplicial distributions, which integrates homotopy-theoretic structures with probability, we introduce event scenarios as a functorial generalization of presheaf-theoretic measurement scenarios and prove their equivalence to bundle scenarios via the Grothendieck construction. We define symmetric monoidal structures on these categories and extend the distribution functor to a stochastic setting, yielding a resource theory that generalizes the presheaf-theoretic notion of simulations. Our main result characterizes convex maps between simplicial distributions in terms of non-contextual distributions on a corresponding mapping scenario, enhancing and extending prior results in categorical quantum foundations.

Related papers

• On the Contractivity of Stochastic Interpolation Flow [1.90365714903665]
  We investigate, a recently introduced framework for high dimensional sampling which bears many similarities to diffusion modeling.<n>We show that for a base distribution and a strongly log-concave target distribution, the flow map is Lipschitz with a sharp constant which matches that of Caffarelli's theorem for optimal transport maps.<n>We are further able to construct Lipschitz transport maps between non-Gaussian distributions, generalizing some recent constructions in the literature on transport methods for establishing functional inequalities.
  arXiv  Detail & Related papers  (2025-04-14T19:10:22Z)
• The geometry of simplicial distributions on suspension scenarios [0.0]
  The geometrical structure of simplicial distributions can be seen as a resource for applications in quantum information theory.<n>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.<n>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.
  arXiv  Detail & Related papers  (2024-12-14T20:47:53Z)
• Compositional Structures in Neural Embedding and Interaction Decompositions [101.40245125955306]
  We describe a basic correspondence between linear algebraic structures within vector embeddings in artificial neural networks. We introduce a characterization of compositional structures in terms of "interaction decompositions" We establish necessary and sufficient conditions for the presence of such structures within the representations of a model.
  arXiv  Detail & Related papers  (2024-07-12T02:39:50Z)
• Twisted simplicial distributions [0.0]
  We leverage the classical theory of simplicial principal bundles, as well as structures on categories of such bundles, to provide powerful computational tools for analyzing twisted distributions. We use these techniques to analyze our key examples: quantum distributions and operator-theoretic polytopes used in the classical simulation of quantum computation.
  arXiv  Detail & Related papers  (2024-03-28T19:43:28Z)
• Distributed Markov Chain Monte Carlo Sampling based on the Alternating Direction Method of Multipliers [143.6249073384419]
  In this paper, we propose a distributed sampling scheme based on the alternating direction method of multipliers. We provide both theoretical guarantees of our algorithm's convergence and experimental evidence of its superiority to the state-of-the-art. In simulation, we deploy our algorithm on linear and logistic regression tasks and illustrate its fast convergence compared to existing gradient-based methods.
  arXiv  Detail & Related papers  (2024-01-29T02:08:40Z)
• Wrapped Distributions on homogeneous Riemannian manifolds [58.720142291102135]
  Control over distributions' properties, such as parameters, symmetry and modality yield a family of flexible distributions. We empirically validate our approach by utilizing our proposed distributions within a variational autoencoder and a latent space network model.
  arXiv  Detail & Related papers  (2022-04-20T21:25:21Z)
• Simplicial quantum contextuality [0.0]
  We introduce a new framework for contextuality based on simplicial sets, models of topological spaces that play a prominent role in homotopy theory. Our approach extends measurement scenarios to consist of spaces (rather than sets) of measurements and outcomes. We present a topologically inspired new proof of Fine's theorem for characterizing noncontextuality in Bell scenarios.
  arXiv  Detail & Related papers  (2022-04-13T22:03:28Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)
• Localisation in quasiperiodic chains: a theory based on convergence of local propagators [68.8204255655161]
  We present a theory of localisation in quasiperiodic chains with nearest-neighbour hoppings, based on the convergence of local propagators. Analysing the convergence of these continued fractions, localisation or its absence can be determined, yielding in turn the critical points and mobility edges. Results are exemplified by analysing the theory for three quasiperiodic models covering a range of behaviour.
  arXiv  Detail & Related papers  (2021-02-18T16:19:52Z)
• Generalization Properties of Optimal Transport GANs with Latent Distribution Learning [52.25145141639159]
  We study how the interplay between the latent distribution and the complexity of the pushforward map affects performance. Motivated by our analysis, we advocate learning the latent distribution as well as the pushforward map within the GAN paradigm.
  arXiv  Detail & Related papers  (2020-07-29T07:31:33Z)
• A structure theorem for generalized-noncontextual ontological models [0.0]
  We use a process-theoretic framework to prove that every generalized-noncontextual ontological model of a tomographically local operational theory has a surprisingly rigid and simple mathematical structure. We extend known results concerning the equivalence of different notions of classicality from prepare-measure scenarios to arbitrary compositional scenarios.
  arXiv  Detail & Related papers  (2020-05-14T17:28:19Z)
• Solvable Criterion for the Contextuality of any Prepare-and-Measure Scenario [0.0]
  An operationally noncontextual ontological model of the quantum statistics associated with the prepare-and-measure scenario is constructed. A mathematical criterion, called unit separability, is formulated as the relevant classicality criterion. We reformulate our results in the framework of generalized probabilistic theories.
  arXiv  Detail & Related papers  (2020-03-13T18:00:05Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.