Entropy Production for Discrete-Time Markov Processes

We study the multiple definitions of the entropy production for discrete-time Markov processes in single systems and composite systems. These definitions have been studied in single systems, but less so in composite systems. With a clear distinction, we review the equivalence condition and the meaning of the multiple definitions and show that all definitions satisfy the important property for the entropy production, such as non-negativity. We also show that the inequalities between total entropy production and marginal entropy production holds for a definition but doesn't for another definition. Furthermore, we verify that fact by calculating entropy productions for Gaussian process and numerically show the result. Finally, we find appropriate use of each definition taking all results into account.