Entanglement Distribution Statistic in Andreev Billiards

We investigate statistical aspects of the entanglement production for open chaotic mesoscopic billiards in contact with superconducting parts, known as Andreev billiards. The complete distributions of concurrence and entanglement of formation are obtained by using the Altland-Zirnbauer symmetry classes of circular ensembles of scattering matrices, which complements previous studies in chaotic universal billiards belonging to other classes of random matrix theory. Our results show a unique and very peculiar behavior: the realization of entanglement in a Andreev billiard always results in non-separable state, regardless of the time reversal symmetry. The analytical calculations are supported by a numerical Monte Carlo simulation.