On limiting distributions of quantum Markov chains. (English) Zbl 1225.81080
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 [D. Aharonov, A. Ambainis, J. Kempe and U. Vazirani, in: Proc. of the 33rd ACM Symposium of Theory of Computing, ACM, New York, NY, USA, 50–59 (2001)].
MSC:
| 81S25 | Quantum stochastic calculus |
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
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