  Nonlocal properties of an ensemble of diagonal random unitary matrices of
order $N^2$ are investigated. The average Schmidt strength of such a bipartite
diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log
N^2$ behavior characteristic to random unitary gates. Entangling power of a
diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum
state $\rho=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained by
reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square
matrix of order $N$. This fact provides a motivation to study the ensemble of
non-hermitian unimodular matrices $A$, with all entries of the same modulus and
random phases and the ensemble of quantum states $\rho$, such that all their
diagonal entries are equal to $1/N$. Such a state is contradiagonal with
respect to the computational basis, in sense that among all unitary equivalent
states it maximizes the entropy copied to the environment due to the coarse
graining process. The first four moments of the squared singular values of the
unimodular ensemble are derived, based on which we conjecture a connection to a
recently studied combinatorial object called the "Borel triangle". This allows
us to find exactly the mean von Neumann entropy for random phase density
matrices and the average entanglement for the corresponding ensemble of
bipartite pure states.
