  Speed limit for classical stochastic Markov processes with discrete states is
studied. We find that a trade-off inequality exists between the speed of the
state transformation and the entropy production. The dynamical activity
determines the time scale and plays a crucial role in the inequality. For
systems with stationary current, a similar trade-off inequality with the
Hatano-Sasa entropy production gives a much better bound on the speed of the
state transformation. Our inequalities contain only physically well-defined
quantities, and thus the physical picture of these inequalities is clear.
