  We derive geometrical bounds on the irreversibility in both quantum and
classical Markovian open systems that satisfy the detailed balance condition.
Using information geometry, we prove that irreversible entropy production is
bounded from below by a modified Wasserstein distance between the initial and
final states, thus strengthening the Clausius inequality in the
reversible-Markov case. The modified metric can be regarded as a discrete-state
generalization of the Wasserstein metric, which has been used to bound
dissipation in continuous-state Langevin systems. Notably, the derived bounds
can be interpreted as the quantum and classical speed limits, implying that the
associated entropy production constrains the minimum time of transforming a
system state. We illustrate the results on several systems and show that a
tighter bound than the Carnot bound for the efficiency of quantum heat engines
can be obtained.
