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Algebraic Phase Theory III: Structural Quantum Codes over Frobenius Rings

Joe Gildea

TL;DR

This work shows that quantum phase, Weyl noncommutativity, and stabiliser codes arise automatically from Frobenius duality in finite Frobenius rings, without relying on Hilbert spaces or analytic structures. It introduces Algebraic Phase Theory (APT) as a framework in which admissible phase data yield a canonical algebraic phase model and a Frobenius phase pairing that encodes all quantum structure. Stabiliser codes correspond to self-orthogonal submodules under this pairing, with CSS codes appearing only as a special splitting case; non-CSS stabilisers naturally arise in general Frobenius rings. Nilpotent and torsion elements in the base ring give intrinsically protected quantum layers, showing that quantisation is a phase-induction phenomenon and that quantum information is fully captured by algebraic phase relations.

Abstract

We develop the quantum component of Algebraic Phase Theory by showing that quantum phase, Weyl noncommutativity, and stabiliser codes arise as unavoidable algebraic consequences of Frobenius duality. Working over finite commutative Frobenius rings, we extract nondegenerate phase pairings, Weyl operator algebras, and quantum stabiliser codes directly from admissible phase data, without assuming Hilbert spaces, analytic inner products, or an externally imposed symplectic structure. Within this framework, quantum state spaces appear as minimal carriers of faithful phase action, and stabiliser codes are identified canonically with self-orthogonal submodules under the Frobenius phase pairing. CSS-type constructions arise only as a special splitting case, while general Frobenius rings admit intrinsically non-CSS stabilisers. Nilpotent and torsion structure in the base ring give rise to algebraically protected quantum layers that are invisible to admissible Weyl-type errors. These results place quantum stabiliser theory within Algebraic Phase Theory: quantisation emerges as algebraic phase induction rather than analytic completion, and quantum structure is information-complete at the level of algebraic phase relations alone. Throughout, we work over finite Frobenius rings, which are precisely the base rings for which admissible phase data become strongly admissible, and in this regime the full quantum formalism is forced by Frobenius duality.

Algebraic Phase Theory III: Structural Quantum Codes over Frobenius Rings

TL;DR

This work shows that quantum phase, Weyl noncommutativity, and stabiliser codes arise automatically from Frobenius duality in finite Frobenius rings, without relying on Hilbert spaces or analytic structures. It introduces Algebraic Phase Theory (APT) as a framework in which admissible phase data yield a canonical algebraic phase model and a Frobenius phase pairing that encodes all quantum structure. Stabiliser codes correspond to self-orthogonal submodules under this pairing, with CSS codes appearing only as a special splitting case; non-CSS stabilisers naturally arise in general Frobenius rings. Nilpotent and torsion elements in the base ring give intrinsically protected quantum layers, showing that quantisation is a phase-induction phenomenon and that quantum information is fully captured by algebraic phase relations.

Abstract

We develop the quantum component of Algebraic Phase Theory by showing that quantum phase, Weyl noncommutativity, and stabiliser codes arise as unavoidable algebraic consequences of Frobenius duality. Working over finite commutative Frobenius rings, we extract nondegenerate phase pairings, Weyl operator algebras, and quantum stabiliser codes directly from admissible phase data, without assuming Hilbert spaces, analytic inner products, or an externally imposed symplectic structure. Within this framework, quantum state spaces appear as minimal carriers of faithful phase action, and stabiliser codes are identified canonically with self-orthogonal submodules under the Frobenius phase pairing. CSS-type constructions arise only as a special splitting case, while general Frobenius rings admit intrinsically non-CSS stabilisers. Nilpotent and torsion structure in the base ring give rise to algebraically protected quantum layers that are invisible to admissible Weyl-type errors. These results place quantum stabiliser theory within Algebraic Phase Theory: quantisation emerges as algebraic phase induction rather than analytic completion, and quantum structure is information-complete at the level of algebraic phase relations alone. Throughout, we work over finite Frobenius rings, which are precisely the base rings for which admissible phase data become strongly admissible, and in this regime the full quantum formalism is forced by Frobenius duality.
Paper Structure (23 sections, 13 theorems, 102 equations)

This paper contains 23 sections, 13 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Structural Background from Algebraic Phase Theory
  3. Algebraic Phase Theory
  4. Frobenius Phase Data and Algebraic Quantum Structure
  5. Algebraic Quantum State Spaces
  6. Tensor extension of the phase pairing
  7. Weyl Operators and the Algebraic Phase Model
  8. Stabiliser Codes and Frobenius Self-Orthogonality
  9. CSS Codes as a Splitting Condition
  10. Quantisation as Algebraic Phase Induction, Invariants, and Equivalence
  11. Nilpotent Protection and Rigidity
  12. Reconstruction from Phase Relations
  13. Symmetry and Normalisers
  14. Examples
  15. $R=\mathbb{F}_2+u\mathbb{F}_2$ with $u^2=0$
  16. ...and 8 more sections

Key Result

Proposition 3.3

The induced pairing on $H_n$ is biadditive, symmetric, monoidal with respect to tensor product, and nondegenerate.

Theorems & Definitions (48)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • ...and 38 more