Diagonal unitary entangling gates and contradiagonal quantum states

Abstract

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 $lnN$, in contrast to the $ln{N}^{2}$ behavior characteristic of random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $\ensuremath{\rho}=A{A}^{\ifmmode\dagger\else\textdagger\fi{}}/{N}^{2}$, 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 $\ensuremath{\rho}$, such that all their diagonal entries are equal to $1/N$. Such a state is contradiagonal with respect to the computational basis, in the 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.