A Mathematical Framework for Transformations of Physical Processes

• URL: http://arxiv.org/abs/2204.04319v2
• Date: Thu, 26 Jan 2023 03:00:25 GMT
• Title: A Mathematical Framework for Transformations of Physical Processes
• Authors: Matt Wilson, Giulio Chiribella
• Abstract summary: We observe that the existence of sequential and parallel composition supermaps in higher order physics can be formalised using enriched category theory. We use the enriched monoidal setting to construct a suitable definition of structure preserving map between higher order physical theories. In a second application we use our definition of structure preserving map to show that categories containing infinite towers of enriched monoidal categories with full and faithful structure preserving maps between them inevitably lead to closed monoidal structures.
• Score: 0.7614628596146599
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We observe that the existence of sequential and parallel composition supermaps in higher order physics can be formalised using enriched category theory. Encouraged by physically relevant examples such as unitary supermaps and layers within higher order causal categories (HOCCs), we treat the modelling of higher order physical theories with enriched monoidal categories in analogy with the modelling of physical theories are with monoidal categories. We use the enriched monoidal setting to construct a suitable definition of structure preserving map between higher order physical theories via the Grothendieck construction. We then show that the convenient feature of currying in higher order physical theories can be seen as a consequence of combining the primitive assumption of the existence of parallel and sequential composition supermaps with an additional feature of linking. In a second application we use our definition of structure preserving map to show that categories containing infinite towers of enriched monoidal categories with full and faithful structure preserving maps between them inevitably lead to closed monoidal structures. The aim of the proposed definitions is to step towards providing a broad framework for the study and comparison of novel causal structures in quantum theory, and, more broadly, a paradigm of physical theory where static and dynamical features are treated in a unified way.

Related papers

• 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)
• Learning Discrete Concepts in Latent Hierarchical Models [73.01229236386148]
  Learning concepts from natural high-dimensional data holds potential in building human-aligned and interpretable machine learning models. We formalize concepts as discrete latent causal variables that are related via a hierarchical causal model. We substantiate our theoretical claims with synthetic data experiments.
  arXiv  Detail & Related papers  (2024-06-01T18:01:03Z)
• Pregeometry, Formal Language and Constructivist Foundations of Physics [0.0]
  We discuss the metaphysics of pregeometric structures, upon which new and existing notions of quantum geometry may find a foundation. We draw attention to evidence suggesting that the framework of formal language, in particular, homotopy type theory, provides the conceptual building blocks for a theory of pregeometry.
  arXiv  Detail & Related papers  (2023-11-07T13:19:29Z)
• Entanglement of Sections: The pushout of entangled and parameterized quantum information [0.0]
  Recently Freedman & Hastings asked for a mathematical theory that would unify quantum entanglement/tensor-structure with parameterized/-structure. We make precise a form of the relevant pushout diagram in monoidal category theory. We show how this model category serves as categorical semantics for the linear-multiplicative fragment of Linear Homotopy Type Theory.
  arXiv  Detail & Related papers  (2023-09-13T18:28:43Z)
• The Generative Programs Framework [0.0]
  We argue that any quantitative physical theory can be represented in the form of a generative program. We suggest that these graphs can be interpreted as encoding relations of ontological priority,' and that ontological priority is a suitable generalisation of causation.
  arXiv  Detail & Related papers  (2023-07-21T00:57:05Z)
• A Category-theoretical Meta-analysis of Definitions of Disentanglement [97.34033555407403]
  Disentangling the factors of variation in data is a fundamental concept in machine learning. This paper presents a meta-analysis of existing definitions of disentanglement.
  arXiv  Detail & Related papers  (2023-05-11T15:24:20Z)
• DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion [66.21290235237808]
  We introduce an energy constrained diffusion model which encodes a batch of instances from a dataset into evolutionary states. We provide rigorous theory that implies closed-form optimal estimates for the pairwise diffusion strength among arbitrary instance pairs. Experiments highlight the wide applicability of our model as a general-purpose encoder backbone with superior performance in various tasks.
  arXiv  Detail & Related papers  (2023-01-23T15:18:54Z)
• Geometry Interaction Knowledge Graph Embeddings [153.69745042757066]
  We propose Geometry Interaction knowledge graph Embeddings (GIE), which learns spatial structures interactively between the Euclidean, hyperbolic and hyperspherical spaces. Our proposed GIE can capture a richer set of relational information, model key inference patterns, and enable expressive semantic matching across entities.
  arXiv  Detail & Related papers  (2022-06-24T08:33:43Z)
• Causality in Higher Order Process Theories [0.7614628596146599]
  We provide an equivalent construction of the HOPT framework from four simple axioms of process-theoretic nature. We then use the HOPT framework to establish connections between foundational features such as causality, determinism and signalling.
  arXiv  Detail & Related papers  (2021-07-30T12:36:12Z)
• A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing [0.0]
  This thesis recast quantum self-testing [MY98,MY04] in operational terms. An input-output process is modelled by a causally structured channel in some fixed theory. implementations are modelled by causally structured dilations formalising hidden side-computations.
  arXiv  Detail & Related papers  (2021-03-03T10:32:34Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)

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.