  We provide upper bound on the maximal rate at which irreversible quantum
dynamics can generate entanglement in a bipartite system. The generator of
irreversible dynamics consists of a Hamiltonian and dissipative terms in
Lindblad form. The relative entropy of entanglement is chosen as a measure of
entanglement in an ancilla-free system. We provide an upper bound on the
entangling rate which has a logarithmic dependence on a dimension of a smaller
system in a bipartite cut. We also investigate the rate of change of quantum
mutual information in an ancilla-assisted system and provide an upper bound
independent of dimension of ancillas.
