Process dimension of classical and non-commutative processes

Publikationstyp
Journal Article
Date Issued
2011-08-19
Sprache
English
Author(s)
Löhr, Wolfgang  
Szkoła, Arleta  
Ay, Nihat  
TORE-URI
http://hdl.handle.net/11420/14557
Journal
Open systems & information dynamics  
Volume
19
Issue
1
Article Number
1250007
Citation
Open Syst. Inf. Dyn. 19 (1): 1250007 (2011-08-19)
Publisher DOI
10.1142/S1230161212500072
Scopus ID
2-s2.0-84859471504
ArXiv ID
1108.3984v1
Publisher
Springer Science + Business Media B.V
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.
Subjects
Mathematics - Dynamical Systems
Mathematics - Dynamical Systems
Mathematical Physics
Mathematics - Mathematical Physics
Mathematics - Probability
Quantum Physics
DDC Class
510: Mathematik
530: Physik