Discrete-time thermodynamic speed limit

As a fundamental thermodynamic principle, speed limits reveal the lower bound of entropy production (EP) required for a system to transition from a given initial state to a final state. While various speed limits have been developed for continuous-time Markov processes, their application to discrete-time Markov chains remains unexplored. In this study, we investigate the speed limits in discrete-time Markov chains, focusing on two types of EP commonly used to measure the irreversibility of a discrete-time process: time-reversed EP and time-backward EP. We find that time-reversed EP satisfies the speed limit for the continuous-time Markov processes, whereas time-backward EP does not. Additionally, for time-reversed EP, we derive practical speed limits applicable to systems driven by cyclic protocols or with unidirectional transitions, where conventional speed limits become meaningless or invalid. We show that these relations also hold for continuous-time Markov processes by taking the time-continuum limit of our results. Finally, we validate our findings through several examples.