Quantum Markov chains (QMCs) on graphs and trees were investigated in connection with many important models arising from quantum statistical mechanics and quantum information. These quantum states generate many important properties such as quantum phase transition and clustering properties. In the present paper, we propose a construction of QMCs associated with an $XX$-Ising model over the comb graph $\mathbb{N}{⊳}_0\mathbb{Z}$. Mainly, we prove that the QMC associated with the disordered phase, enjoys a clustering property.

We established a continuous-time framework for entangled quantum Markov chains derived from classical continuous-time Markov processes. Unlike previous studies focused on discrete-time dynamics, we demonstrated that a robust mixing property emerges directly from the structural architecture of the underlying continuous-time process. Our analysis showed that inhomogeneous entangled chains exhibited path-independent convergence to a limiting distribution determined by asymptotic temporal parameters. The results motivated future study of these entangled Markov chains in connection with open quantum system dynamics and quantum master equations.