  We analyze the optimal basis for generating the maximum relative entropy of
quantum coherence by an arbitrary gate on a two-qubit system. The optimal basis
is not unique, and the high quantum coherence generating gates are also
typically high entanglement generating ones and vice versa. However, the
profile of the relative frequencies of Haar random unitaries generating
different amounts of entanglement for a fixed amount of quantum coherence is
different from the one in which the roles of entanglement and quantum coherence
are reversed, although both follow the beta distribution.
