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Intrinsic Geometry of Operational Contexts: A Riemannian-Style Framework for Quantum Channels

Kazuyuki Yoshida

TL;DR

The paper proposes an intrinsic N--Q--S geometry of operational contexts, positing contexts, channels, and self-preservation as primary constructs and deriving metric and curvature notions from their interactions. It develops a three-layer architecture (N-layer openness, Q-layer GKLS channels with a Doeblin spectral gap, and S-layer self-preservation functionals), whose Hessians yield information-geometric metrics on internal charges and whose questioning loops define a non-commutative curvature. Discrete context graphs give rise to a context Laplacian and a global self-preservation action, with continuum limits recovering Laplace–Beltrami or d'Alembert operators and, in favorable regimes, Einstein-type equations for emergent gravity and gauge structures as natural charts of the geometry. The framework unifies classical Riemannian, quantum Fisher, and holonomy-based gauge concepts within a single operational geometry, offering a route to emergent space–time and interactions from information-theoretic and contextual considerations.

Abstract

We propose an intrinsic geometric framework on the space of operational contexts, specified by channels, stationary states, and self-preservation functionals. Each context C carries a pointer algebra, internal charges, and a self-consistent configuration minimizing a self-preservation functional. The Hessian of this functional yields an intrinsic metric on charge space, while non-commutative questioning loops dN -> dPhi -> d rho^circ define a notion of curvature. In suitable regimes, this N-Q-S geometry reduces to familiar Fisher-type information metrics and admits charts that resemble Riemannian or Lorentzian space-times. We outline how gauge symmetries and gravitational dynamics can be interpreted as holonomies and consistency conditions in this context geometry.

Intrinsic Geometry of Operational Contexts: A Riemannian-Style Framework for Quantum Channels

TL;DR

The paper proposes an intrinsic N--Q--S geometry of operational contexts, positing contexts, channels, and self-preservation as primary constructs and deriving metric and curvature notions from their interactions. It develops a three-layer architecture (N-layer openness, Q-layer GKLS channels with a Doeblin spectral gap, and S-layer self-preservation functionals), whose Hessians yield information-geometric metrics on internal charges and whose questioning loops define a non-commutative curvature. Discrete context graphs give rise to a context Laplacian and a global self-preservation action, with continuum limits recovering Laplace–Beltrami or d'Alembert operators and, in favorable regimes, Einstein-type equations for emergent gravity and gauge structures as natural charts of the geometry. The framework unifies classical Riemannian, quantum Fisher, and holonomy-based gauge concepts within a single operational geometry, offering a route to emergent space–time and interactions from information-theoretic and contextual considerations.

Abstract

We propose an intrinsic geometric framework on the space of operational contexts, specified by channels, stationary states, and self-preservation functionals. Each context C carries a pointer algebra, internal charges, and a self-consistent configuration minimizing a self-preservation functional. The Hessian of this functional yields an intrinsic metric on charge space, while non-commutative questioning loops dN -> dPhi -> d rho^circ define a notion of curvature. In suitable regimes, this N-Q-S geometry reduces to familiar Fisher-type information metrics and admits charts that resemble Riemannian or Lorentzian space-times. We outline how gauge symmetries and gravitational dynamics can be interpreted as holonomies and consistency conditions in this context geometry.

Paper Structure

This paper contains 49 sections, 8 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context space and pointer algebras
  3. Q-layer: channels, fixed points, and mixing
  4. Primitive channels and stationary states
  5. Doeblin-type mixing and spectral gap
  6. Poisson-type equation and sensitivity of fixed points
  7. S-layer: self-preservation functionals and intrinsic metric
  8. Self-preservation functional and self-consistency
  9. Intrinsic metric from the Hessian
  10. Relation to Fisher information in simple models
  11. Combining gap and metric: a basic response inequality
  12. N-layer: context graph, ghost couplings, and Laplacians
  13. Context graph and ghost couplings
  14. Global self-preservation action and discrete field equations
  15. Toward Einstein-type equations in continuum limits
  16. ...and 34 more sections

Key Result

Theorem 1.2

Assume an N--Q--S architecture as in Definition def:NQS-architecture with a finite context graph, GKLS generators with Doeblin-certified spectral gaps, and smooth self-preservation functionals. Then:

Figures (1)

  • Figure 1: Schematic N--Q--S architecture and sensitivity chain $dN \to d\mathcal{L}_C \to d\rho_C^\circ \to dq(C) \to d^2\mathcal{F}_C$.

Theorems & Definitions (41)

  • Definition 1.1: N--Q--S architecture
  • Theorem 1.2: Main structural result, informal
  • Definition 2.1: Context space
  • Definition 2.2: Pointer algebra and internal symmetry
  • Remark 2.3
  • Definition 3.1: NQS channel family
  • Definition 3.2: Primitive channel
  • Definition 3.3: Doeblin-type minorization
  • Proposition 3.4: Spectral gap from Doeblin-type minorization
  • Remark 3.5
  • ...and 31 more