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Nonlocal properties of an ensemble of diagonal random unitary matrices oforder $N^2$ are investigated. The average Schmidt strength of such a bipartitediagonal quantum gate is shown to scale as $\log N$, in contrast to the $\logN^2$ behavior characteristic to random unitary gates. Entangling power of adiagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantumstate $\rho=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained byreshaping the vector of diagonal elements of $U$ of length $N^2$ into a squarematrix of order $N$. This fact provides a motivation to study the ensemble ofnon-hermitian unimodular matrices $A$, with all entries of the same modulus andrandom phases and the ensemble of quantum states $\rho$, such that all theirdiagonal entries are equal to $1/N$. Such a state is contradiagonal withrespect to the computational basis, in sense that among all unitary equivalentstates it maximizes the entropy copied to the environment due to the coarsegraining process. The first four moments of the squared singular values of theunimodular ensemble are derived, based on which we conjecture a connection to arecently studied combinatorial object called the "Borel triangle". This allowsus to find exactly the mean von Neumann entropy for random phase densitymatrices and the average entanglement for the corresponding ensemble ofbipartite pure states.

Diagonal unitary entangling gates and contradiagonal quantum states