{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:54:15Z","timestamp":1753883655575,"version":"3.41.2"},"reference-count":33,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2011,8,25]],"date-time":"2011-08-25T00:00:00Z","timestamp":1314230400000},"content-version":"vor","delay-in-days":236,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1005564"],"award-info":[{"award-number":["CCF-1005564"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>In a quantum Markov chain, the temporal succession of states\nis modeled by the repeated action of a \u201cbistochastic quantum operation\u201d on\nthe density matrix of a quantum system. Based on this conceptual framework,\nwe derive some new results concerning the evolution of a quantum system,\nincluding its long\u2010term behavior. Among our findings is the fact that\nthe Ces\u00e0ro limit of any quantum Markov chain always exists and equals\nthe orthogonal projection of the initial state upon the eigenspace of the\nunit eigenvalue of the bistochastic quantum operation. Moreover, if the\nunit eigenvalue is the only eigenvalue on the unit circle, then the quantum\nMarkov chain converges in the conventional sense to the said orthogonal\nprojection. As a corollary, we offer a new derivation of the classic result\ndescribing limiting distributions of unitary quantum walks on finite graphs\n(Aharonov et al., 2001).<\/jats:p>","DOI":"10.1155\/2011\/740816","type":"journal-article","created":{"date-parts":[[2011,8,25]],"date-time":"2011-08-25T19:00:48Z","timestamp":1314298848000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["On Limiting Distributions of Quantum Markov Chains"],"prefix":"10.1155","volume":"2011","author":[{"given":"Chaobin","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nelson","family":"Petulante","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2011,8,25]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"AharonovD. AmbainisA. KempeJ. andVaziraniU. Quantum walks on graphs Proceedings of the 33rd ACM Symposium on Theory of Computing 2001 New York NY USA ACM 50\u201359 https:\/\/doi.org\/10.1145\/380752.380758 2120300.","key":"e_1_2_8_1_2","DOI":"10.1145\/380752.380758"},{"doi-asserted-by":"publisher","key":"e_1_2_8_2_2","DOI":"10.1142\/S0219749903000383"},{"doi-asserted-by":"crossref","unstructured":"AmbainisA. BachE. NayakA. VishwanathA. andWatrousJ. One-dimensional quantum walks Proceedings of the 33d Annual ACM Symposium on Theory of Computing 2001 New York NY USA ACM 37\u201349 https:\/\/doi.org\/10.1145\/380752.380757 2120299.","key":"e_1_2_8_3_2","DOI":"10.1145\/380752.380757"},{"doi-asserted-by":"publisher","key":"e_1_2_8_4_2","DOI":"10.1023\/A:1019609420309"},{"doi-asserted-by":"publisher","key":"e_1_2_8_5_2","DOI":"10.1103\/PhysRevA.58.915"},{"doi-asserted-by":"publisher","key":"e_1_2_8_6_2","DOI":"10.1080\/00107151031000110776"},{"doi-asserted-by":"publisher","key":"e_1_2_8_7_2","DOI":"10.1017\/S0960129507006354"},{"doi-asserted-by":"publisher","key":"e_1_2_8_8_2","DOI":"10.1007\/978-3-540-69365-9_7"},{"unstructured":"NayakA.andVishwanathA. 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