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

TL;DR

This paper introduces and analyzes time-inhomogeneous quantum Markov chains with decoherence on finite spaces, exploring their equilibrium properties and phase transitions related to temperature and decoherence strength.

Contribution

It provides the first rigorous analysis of ergodic and phase transition behaviors in quantum Markov chains with time-inhomogeneity and decoherence.

Findings

Quantum Markov chains are ergodic for small ta in high temperature.

Multiple limiting distributions occur at large ta.

Phase transition is expected at critical point ta_c=1.

Abstract

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