Process Dimension of Classical and Non-Commutative Processes

Details Process Dimension of Classical and Non-Commutative Processes Process Dimension of Classical and Non-Commutative Processes

2011-08-19

arxiv.org/abs/1108.3984v1

Open Syst. Inf. Dyn. 19(1), 2012

Mathematics

Dynamical Systems

8 pages

Scientific paper

10.1142/S1230161212500072

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.

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