Unitary quantum gates, perfect entanglers, and unistochastic maps

Abstract

Nonlocal 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 behavior. 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