  We discuss upper bounds on the rate at which unitary evolution governed by a
non-local Hamiltonian can generate entanglement in a bipartite system. Given a
bipartite Hamiltonian H coupling two finite dimensional particles A and B, the
entangling rate is shown to be upper bounded by c*log(d)*norm(H), where d is
the smallest dimension of the interacting particles, norm(H) is the operator
norm of H, and c is a constant close to 1. Under certain restrictions on the
initial state we prove analogous upper bound for the ancilla-assisted
entangling rate with a constant c that does not depend upon dimensions of local
ancillas. The restriction is that the initial state has at most two distinct
Schmidt coefficients (each coefficient may have arbitrarily large
multiplicity). Our proof is based on analysis of a mixing rate -- a functional
measuring how fast entropy can be produced if one mixes a time-independent
state with a state evolving unitarily.
