Model reduction via lumping in classical and quantum systems

When using models of complex networks or large systems, a typical problem arises from their high dimensionality, which usually makes them computationally intractable. To address this issue, several model reduction approaches have been proposed in the literature. The first part of this work aims to analyze the existing literature on the model reduction of discrete-time Markov chains, utilizing the lumpability concepts formulated by Buchholz. In particular, we will consider the WSP framework proposed by Michel-Siegle, highlighting its strong relationship with the algebraic reduction framework for hidden Markov models developed by Grigoletto-Ticozzi, and deriving the results from a system-theoretic point of view. The second part is devoted to extending several concepts and results derived in the first part to discrete-time quantum systems. For instance, we will introduce the definition of quantum ordinary lumpability and its characterizations.