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

TL;DR

This paper introduces a Riemannian-style geometric framework for quantum channels and operational contexts, linking information metrics, curvature, and gauge symmetries to quantum information and gravitational concepts.

Contribution

It develops an intrinsic geometric structure on quantum operational contexts, connecting charge spaces, curvature, and gauge symmetries within a unified framework.

Findings

Defines an intrinsic metric from the Hessian of a self-preservation functional

Shows reduction to Fisher-type metrics in certain regimes

Interprets gauge symmetries and gravity as holonomies in the geometry

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.