Defect Cocycles and the Structure of Finite Process Monoids

TL;DR

This paper investigates the structure of finite families of positive subunital maps in ordered effect spaces, proving defect annihilation under iteration and establishing bounds on stabilization indices, with implications for quantum process theories.

Contribution

It introduces a spectral-free, order-theoretic approach to analyze defect behavior in positive maps, providing explicit bounds and conditions for unitality in finite process monoids.

Findings

Defects are eventually annihilated under iteration in finite families.

All maps are unital under a persistence hypothesis.

Dimension-dependent bounds on stabilization indices are established.

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.