Time-inhomogeneous Quantum Markov Chains with Decoherence on Finite State Spaces

Abstract

We introduce and study time-inhomogeneous quantum Markov chains with parameter ζ ≥ 0 and decoherence parameter 0 ≤ p ≤ 1 on finite spaces and their large scale equilibrium properties. Here ζ resembles the inverse temperature in the annealing random process and p is the decoherence strength of the quantum system. Numerical evaluations show that if ζ is small, then quantum Markov chain is ergodic for all 0 < p ≤ 1 and if ζ is large, then it has multiple limiting distributions for all 0 < p ≤ 1. In this paper, we prove the ergodic property in the high temperature region 0 ≤ ζ ≤ 1. We expect that the phase transition occurs at the critical point ζ c = 1. For coherence case p = 0, a critical behavior of periodicity also appears at critical point ζ o = 2.