On limiting distributions of quantum Markov chains

Abstract

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 Ces$\grave{a}$ro 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 \cite{AAKV01}.

References in corpus (8)

• Decoherence in quantum walks - a review
• A quantum version of Wielandt's inequality
• Random Quantum Operations
• Decoherence in one-dimensional Quantum Walk
• Quantumness in decoherent quantum walk using measurement-induced disturbance
• Asymptotic evolution of quantum walks on the $N$-cycle subject to decoherence on both the coin and position degrees of freedom
• Quantum random walks on the $N$-cycle subject to decoherence on the coin degree of freedom
• Localization and Fractality in Inhomogeneous Quantum Walks with Self-Duality

Cited by in corpus (6)

• Quantum walks: a comprehensive review
• On a class of quantum channels, open random walks and recurrence
• Open quantum random walks: ergodicity, hitting times, gambler's ruin and potential theory
• Open Quantum Random Walks and Quantum Markov chains on Trees I: Phase transitions
• Unitary conjugation channels with continuous random phases
• A quantization procedure based on completely positive maps and Markov operators