2015/04/21 by Carlos F. Lardizabal, Lardizabal, Carlos F., Rafael Rigão Souza +1
Computer Science · Mathematics · #FOS: Mathematics #FOS: Physical sciences #Graph theory and applications #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1504.05398
openalex publication_date 2015/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by a model presented by S. Gudder, we study a quantum generalization of Markov chains and discuss the relation between these maps and open quantum random walks, a class of quantum channels described by S. Attal et al. We consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Zúñiga, we define a notion of ergodicity for nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a consequence we obtain a quantum version of the classical probability result concerning the behavior of the columns (or rows) of the iterates of a stochastic matrix induced by a finite, irreducible, aperiodic Markov chain. We are also able to relate the ergodic property presented here with the notions of weak and uniform ergodicity known in the literature of noncommutative L1-spaces.