Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Defect Cocycles and the Structure of Finite Process Monoids

Paolo Vella

TL;DR

The paper introduces the defect cocycle for positive subunital maps on ordered effect spaces and shows that, in any finite composition-closed family, each map achieves defect annihilation after at most $n_T\le |A|$ iterations, providing a structural reason finite repertoires cannot sustain information loss. Under a persistence hypothesis, all maps in such a family become unital, linking losslessness to normalization preservation. For finite-dimensional quantum operations, the authors prove the sharp bound $n_T\le d$ (dimension of the Hilbert space), with a shift channel realizing equality. The analysis extends to infinite dimensions via asymptotic defects, corner-nilpotency, and discrete Lyapunov spectra, and develops both atomic-center and diffuse-composition criteria for stabilization under parallel and sequential composition. These results illuminate why finite operational repertoires in process theories inherently stabilize and prevent lasting leakage, with implications for quantum foundations and categorical probability.

Abstract

We study positive subunital maps on ordered effect spaces and introduce the defect $d(T) = u - T(u)$, which satisfies a cocycle identity under composition. Using only this identity and elementary order-theoretic arguments -- requiring no spectral decomposition or dimension-dependent techniques -- we prove that in any finite composition-closed family of positive subunital maps, defects are eventually annihilated under iteration (Theorem 4.1), with an explicit bound linear in the family size. Under a persistence hypothesis (nonzero positive elements map to nonzero positive elements), we establish that all maps in such families must be unital. For completely positive maps on finite-dimensional matrix algebras, we then prove a sharp dimension-dependent bound: the stabilization index satisfies $n_T \le d$ where $d$ is the Hilbert space dimension, independent of the family size. This bound is achieved by a shift channel construction. These results provide a structural explanation for why finite operational repertoires in process theories cannot sustain systematic information loss, with applications to quantum foundations and categorical probability.

Defect Cocycles and the Structure of Finite Process Monoids

TL;DR

The paper introduces the defect cocycle for positive subunital maps on ordered effect spaces and shows that, in any finite composition-closed family, each map achieves defect annihilation after at most iterations, providing a structural reason finite repertoires cannot sustain information loss. Under a persistence hypothesis, all maps in such a family become unital, linking losslessness to normalization preservation. For finite-dimensional quantum operations, the authors prove the sharp bound (dimension of the Hilbert space), with a shift channel realizing equality. The analysis extends to infinite dimensions via asymptotic defects, corner-nilpotency, and discrete Lyapunov spectra, and develops both atomic-center and diffuse-composition criteria for stabilization under parallel and sequential composition. These results illuminate why finite operational repertoires in process theories inherently stabilize and prevent lasting leakage, with implications for quantum foundations and categorical probability.

Abstract

We study positive subunital maps on ordered effect spaces and introduce the defect , which satisfies a cocycle identity under composition. Using only this identity and elementary order-theoretic arguments -- requiring no spectral decomposition or dimension-dependent techniques -- we prove that in any finite composition-closed family of positive subunital maps, defects are eventually annihilated under iteration (Theorem 4.1), with an explicit bound linear in the family size. Under a persistence hypothesis (nonzero positive elements map to nonzero positive elements), we establish that all maps in such families must be unital. For completely positive maps on finite-dimensional matrix algebras, we then prove a sharp dimension-dependent bound: the stabilization index satisfies where is the Hilbert space dimension, independent of the family size. This bound is achieved by a shift channel construction. These results provide a structural explanation for why finite operational repertoires in process theories cannot sustain systematic information loss, with applications to quantum foundations and categorical probability.
Paper Structure (19 sections, 94 theorems, 148 equations)

This paper contains 19 sections, 94 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Clarification: finite composition-closed families.
  3. Related work.
  4. Outline.
  5. Preliminaries
  6. The Defect Cocycle
  7. Main Results
  8. Examples and Counterexamples
  9. Parallel Composition
  10. Asymptotic Defects in Infinite Dimensions
  11. Type I with atomic center.
  12. Finite factors with finite-index structure.
  13. Hyperfinite II$_1$.
  14. Dimension-Dependent Bounds for Quantum Operations
  15. Proof idea.
  16. ...and 4 more sections

Key Result

Lemma 2.6

If $E_+$ is pointed and $x_1, \ldots, x_m \in E_+$ satisfy $x_1 + \cdots + x_m = 0$, then $x_i = 0$ for all $i$. Equivalently: $E_+$ is pointed if and only if for all $x, y \ge 0$, $x + y = 0$ implies $x = y = 0$.

Theorems & Definitions (263)

  • Definition 2.1: Ordered effect space
  • Definition 2.2: Pointed cone
  • Remark 2.3: Relationship between pointedness and antisymmetry
  • Remark 2.4
  • Remark 2.5: Why pointedness is necessary
  • Lemma 2.6: Pointed cone cancellation
  • proof
  • Definition 2.7: Positive and subunital maps
  • Lemma 2.8: Monotonicity
  • proof
  • ...and 253 more