{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T15:45:58Z","timestamp":1648741558439},"reference-count":9,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Open Syst. Inf. Dyn."],"published-print":{"date-parts":[[2012,3]]},"abstract":"<jats:p> 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 one 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. <\/jats:p>","DOI":"10.1142\/s1230161212500072","type":"journal-article","created":{"date-parts":[[2012,3,15]],"date-time":"2012-03-15T06:59:34Z","timestamp":1331794774000},"page":"1250007","source":"Crossref","is-referenced-by-count":0,"title":["Process Dimension of Classical and Non-Commutative Processes"],"prefix":"10.1142","volume":"19","author":[{"given":"Wolfgang","family":"L\u00f6hr","sequence":"first","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany"},{"name":"Universit\u00e4t Duisburg-Essen, Universit\u00e4tsstra\u00dfe 2, 45141 Essen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arleta","family":"Szko\u0142a","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nihat","family":"Ay","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,4,8]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02520-8"},{"key":"rf2","volume-title":"Lectures on Analysis, Volume II (Representation Theory)","author":"Choquet G.","year":"1969"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.63.105"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/BF02099178"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/BF00668821"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177700000"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1162\/089976600300015411"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.3390\/e110300385"},{"key":"rf10","volume-title":"Statistical Mechanics, Rigorous Results","author":"Ruelle D.","year":"1969"}],"container-title":["Open Systems &amp; Information Dynamics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1230161212500072","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T17:26:10Z","timestamp":1565112370000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1230161212500072"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3]]},"references-count":9,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2012,4,8]]},"published-print":{"date-parts":[[2012,3]]}},"alternative-id":["10.1142\/S1230161212500072"],"URL":"https:\/\/doi.org\/10.1142\/s1230161212500072","relation":{},"ISSN":["1230-1612","1793-7191"],"issn-type":[{"value":"1230-1612","type":"print"},{"value":"1793-7191","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,3]]}}}