Correlated Markov Quantum Walks

A. Romanelli

Ameur Dhahri, Chul Ki Ko, Hyun Jae Yoo

Diego de Falco, Dario Tamascelli

Abdessatar Souissi, Abdessatar Barhoumi

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.

Ameur Dhahri, Chul Ki Ko, Hyun Jae Yoo

In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. We then discuss the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. The quantum Markov chain was introduced by Accardi by using transition expectation, and the construction of the quantum Markov chains for the open quantum random walks was introduced by Dhahri and Mukhamedov. Here we construct nonhomogeneous quantum Markov chains for the open quantum random walks so that we can naturally express the evolution of any open quantum random walk by the corresponding quantum Markov chain. We provide with some examples. In particular, we show that the classical Markov chains are reconstructed as quantum Markov chains as well and the reducibility and irreducibility properties of classical Markov chains can be investigated in the language of quantum Markov chains.

Diego de Falco, Dario Tamascelli, Centro Interdipartimentale +1 more