On Limiting Distributions of Quantum Markov Chains
In a quantum Markov chain, the temporal succession of states is modeled by the repeated action of a “bistochastic quantum operation” on the density matrix of a quantum system. Based on this conceptual framework, we derive some new results concerning the evolution of a quantum system, including its l...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2011/740816 |
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| author | Chaobin Liu Nelson Petulante |
| author_facet | Chaobin Liu Nelson Petulante |
| author_sort | Chaobin Liu |
| collection | DOAJ |
| description | In a quantum Markov chain, the temporal succession of states
is modeled by the repeated action of a “bistochastic quantum operation” on
the density matrix of a quantum system. Based on this conceptual framework,
we derive some new results concerning the evolution of a quantum system,
including its long-term behavior. Among our findings is the fact that
the Cesàro limit of any quantum Markov chain always exists and equals
the orthogonal projection of the initial state upon the eigenspace of the
unit eigenvalue of the bistochastic quantum operation. Moreover, if the
unit eigenvalue is the only eigenvalue on the unit circle, then the quantum
Markov chain converges in the conventional sense to the said orthogonal
projection. As a corollary, we offer a new derivation of the classic result
describing limiting distributions of unitary quantum walks on finite graphs
(Aharonov et al., 2001). |
| format | Article |
| id | doaj-art-ac4f1228095a4b2f93a8d18477a58060 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-ac4f1228095a4b2f93a8d18477a580602025-02-03T01:22:39ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252011-01-01201110.1155/2011/740816740816On Limiting Distributions of Quantum Markov ChainsChaobin Liu0Nelson Petulante1Department of Mathematics, Bowie State University, 14000 Jericho Park Road, Bowie, MD 20715, USADepartment of Mathematics, Bowie State University, 14000 Jericho Park Road, Bowie, MD 20715, USAIn a quantum Markov chain, the temporal succession of states is modeled by the repeated action of a “bistochastic quantum operation” on the density matrix of a quantum system. Based on this conceptual framework, we derive some new results concerning the evolution of a quantum system, including its long-term behavior. Among our findings is the fact that the Cesàro limit of any quantum Markov chain always exists and equals the orthogonal projection of the initial state upon the eigenspace of the unit eigenvalue of the bistochastic quantum operation. Moreover, if the unit eigenvalue is the only eigenvalue on the unit circle, then the quantum Markov chain converges in the conventional sense to the said orthogonal projection. As a corollary, we offer a new derivation of the classic result describing limiting distributions of unitary quantum walks on finite graphs (Aharonov et al., 2001).http://dx.doi.org/10.1155/2011/740816 |
| spellingShingle | Chaobin Liu Nelson Petulante On Limiting Distributions of Quantum Markov Chains International Journal of Mathematics and Mathematical Sciences |
| title | On Limiting Distributions of Quantum Markov Chains |
| title_full | On Limiting Distributions of Quantum Markov Chains |
| title_fullStr | On Limiting Distributions of Quantum Markov Chains |
| title_full_unstemmed | On Limiting Distributions of Quantum Markov Chains |
| title_short | On Limiting Distributions of Quantum Markov Chains |
| title_sort | on limiting distributions of quantum markov chains |
| url | http://dx.doi.org/10.1155/2011/740816 |
| work_keys_str_mv | AT chaobinliu onlimitingdistributionsofquantummarkovchains AT nelsonpetulante onlimitingdistributionsofquantummarkovchains |