Symmetric Group Action on Quantum Markov Chains on Cayley Trees and Open Quantum Walks Dynamics

This paper introduces a symmetric group dynamics on quantum Markov chains on Cayley trees (QMCCT). By applying symmetric group actions, we establish a novel algebraic perspective that enables the systematic study of local correlation functions in QMCCT. This approach provides a powerful tool for investigating phase transitions and characterizing ergodic properties in these systems. We further extend this framework to explore QMCCT associated with Open Quantum Random Walks (OQRWs). The integration of symmetric group dynamics allows for a detailed analysis of the interplay between group-theoretic symmetries and the stochastic processes underlying QMCCT, particularly in the context of OQRWs. Our results demonstrate the potential of this approach to uncover new insights into the structure and behaviour of quantum Markov chains.