Entanglement in random pure states: Spectral density and average von Neumann entropy

Quantum entanglement plays a crucial role in quantum information, quantum teleportation and quantum computation. The information about the entanglement content between subsystems of the composite system is encoded in the Schmidt eigenvalues. We derive here closed expressions for the spectral density of Schmidt eigenvalues for all three invariant classes of random matrix ensembles. We also obtain exact results for average von Neumann entropy. We find that maximum average entanglement is achieved if the system belongs to the symplectic invariant class.