  Non-local properties of ensembles of quantum gates induced by the Haar
measure on the unitary group are investigated. We analyze the entropy of
entanglement of a unitary matrix U equal to the Shannon entropy of the vector
of singular values of the reshuffled matrix. Averaging the entropy over the
Haar measure on U(N^2) we find its asymptotic behaviour. For two--qubit quantum
gates we derive the induced probability distribution of the interaction content
and show that the relative volume of the set of perfect entanglers reads 8/3
\pi \approx 0.85. We establish explicit conditions under which a given
one-qubit bistochastic map is unistochastic, so it can be obtained by partial
trace over a one--qubit environment initially prepared in the maximally mixed
state.
