Wasserstein distance in speed limit inequalities for Markov jump processes

The role of the Wasserstein distance in the thermodynamic speed limit inequalities for Markov jump processes is investigated. We elucidate the nature of the Wasserstein distance in the thermodynamic speed limit inequality from three different perspectives by resolving three remaining problems. First, we derive a unified speed limit inequality for a general weighted graph, which reproduces both the conventional speed limit inequality and the trade-off relation between current and entropy production as its special case. Second, we treat the setting where the tightest bound with the Wasserstein distance has not yet been obtained and investigate why such a bound is out of reach. Third, we compare the speed limit inequalities for Markov jump processes with the L1-Wasserstein distance and for overdamped Langevin systems with the L2-Wasserstein distance, and argue that these two have different origins, despite their apparent similarity.