Typical entanglement

Let a pure state vertical bar psi > be chosen randomly in an NM-dimensional Hilbert space, and consider the reduced density matrix rho(A) of an N-dimensional subsystem. The bipartite entanglement properties of vertical bar psi > are encoded in the spectrum of rho(A). By means of a saddle point method and using a Coulomb gas model for the eigenvalues, we obtain the typical spectrum of reduced density matrices. We consider the cases of an unbiased ensemble of pure states and of a fixed value of the purity. We finally obtain the eigenvalue distribution by using a statistical mechanics approach based on the introduction of a partition function.