Process Dimension of Classical and Non-Commutative Processes

TL;DR

This paper explores the process dimension in classical and non-commutative processes, establishing its properties and relationship with causal states, thereby extending complexity measures to non-commutative stochastic processes.

Contribution

It introduces a non-commutative generalization of observable operator models and analyzes the process dimension as a complexity measure in this broader context.

Findings

Proves lower semi-continuity of process dimension.

Derives an ergodic decomposition formula.

Shows topological statistical complexity bounds process dimension.

Abstract

We treat observable operator models (OOM) and their non-commutative generalisation, which we call NC-OOMs. A natural characteristic of a stochastic process in the context of classical OOM theory is the process dimension. We investigate its properties within the more general formulation, which allows to consider process dimension as a measure of complexity of non-commutative processes: We prove lower semi-continuity, and derive an ergodic decomposition formula. Further, we obtain results on the close relationship between the canonical OOM and the concept of causal states which underlies the definition of statistical complexity. In particular, the topological statistical complexity, i.e. the logarithm of the number of causal states, turns out to be an upper bound to the logarithm of process dimension.