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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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). |
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| ISSN: | 0161-1712 1687-0425 |