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Non-local properties of ensembles of quantum gates induced by the Haarmeasure on the unitary group are investigated. We analyze the entropy ofentanglement of a unitary matrix U equal to the Shannon entropy of the vectorof singular values of the reshuffled matrix. Averaging the entropy over theHaar measure on U(N^2) we find its asymptotic behaviour. For two--qubit quantumgates we derive the induced probability distribution of the interaction contentand 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 givenone-qubit bistochastic map is unistochastic, so it can be obtained by partialtrace over a one--qubit environment initially prepared in the maximally mixedstate.

Unitary quantum gates, perfect entanglers and unistochastic maps